TY - INPR U1 - Preprint A1 - Thiele, Michael A1 - Tscherpel, Tabea T1 - Asymptotic preserving discretisation and error estimates for compressible gas mixture models N2 - We consider a compressible mixture model for binary fluids with two velocities in one space dimension in the low-Mach, high-friction regime. Building on the framework in [Egger, Giesselmann 2023], we establish stability with respect to the data. Furthermore, we analyse a high-order, energy-consistent numerical discretisation. In particular, we prove asymptotic-preserving error estimates for a first-order time discretisation combined with a spatial discretisation of arbitrary order, under suitable regularity assumptions. The scheme combines a modified continuous Petrov-Galerkin discretisation in time, introduced in [Giesselmann, Karsai, Tscherpel 2025] for a general class of nonlinear port-Hamiltonian systems, with a mixed finite element discretisation in space, both of arbitrary order. By developing and using an approximation operator adapted to the nonlinear structure of the model, we obtain optimal-order error estimates in the spatial discretisation parameter. Our results also apply to a one-component fluid with friction, extending existing schemes and error estimates to higher-order spatial discretisations. AB - We consider a compressible mixture model for binary fluids with two velocities in one space dimension in the low-Mach, high-friction regime. Building on the framework in [Egger, Giesselmann 2023], we establish stability with respect to the data. Furthermore, we analyse a high-order, energy-consistent numerical discretisation. In particular, we prove asymptotic-preserving error estimates for a first-order time discretisation combined with a spatial discretisation of arbitrary order, under suitable regularity assumptions. The scheme combines a modified continuous Petrov-Galerkin discretisation in time, introduced in [Giesselmann, Karsai, Tscherpel 2025] for a general class of nonlinear port-Hamiltonian systems, with a mixed finite element discretisation in space, both of arbitrary order. By developing and using an approximation operator adapted to the nonlinear structure of the model, we obtain optimal-order error estimates in the spatial discretisation parameter. Our results also apply to a one-component fluid with friction, extending existing schemes and error estimates to higher-order spatial discretisations. KW - continuous Petrov-Galerkin KW - a priori error estimates KW - port-Hamiltonian KW - mixtures KW - energy-consistency Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Hante, Falk A1 - Hernandez, Martin A1 - Leugering, Günter A1 - Schmidt, Martin A1 - Zuazua, Enrique T1 - Domain Decomposition for Gas Network Control N2 - We present domain decomposition techniques for efficient and scalable simulation and optimization of partial differential equations on network domains. Motivated by transient gas network applications, the state-of-the-art of space-and-time decompositions for systems derived from Euler's equations are presented. Moreover, combinatorial aspects of switching valves are addressed in a decomposition framework for mixed-integer and ODE-constrained problems and a convergence analysis is presented. Motivated by stochastic approximation, we also propose and explore randomized decomposition approaches. Some of these advanced solution strategies can be combined and regarded as variants of certain penalty alternating direction methods. We present the potential applications of these techniques by summarizing numerical results from the literature for transient GasLib instances modeling a realistic gas transport network. AB - We present domain decomposition techniques for efficient and scalable simulation and optimization of partial differential equations on network domains. Motivated by transient gas network applications, the state-of-the-art of space-and-time decompositions for systems derived from Euler's equations are presented. Moreover, combinatorial aspects of switching valves are addressed in a decomposition framework for mixed-integer and ODE-constrained problems and a convergence analysis is presented. Motivated by stochastic approximation, we also propose and explore randomized decomposition approaches. Some of these advanced solution strategies can be combined and regarded as variants of certain penalty alternating direction methods. We present the potential applications of these techniques by summarizing numerical results from the literature for transient GasLib instances modeling a realistic gas transport network. Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Breitkopf, Jannik A1 - Ulbrich, Stefan T1 - An Adjoint Calculus for Optimal Control of Entropy Solutions of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws with Shocks N2 - We derive an adjoint gradient representation for the objective functional of an optimal control problem of the unique entropy solution to the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation or balance laws. The GRP is an initial value problem where the initial state is piecewise C^1 with exactly one discontinuity. In a recent work, differentiability properties of the solution operator of the GRP were derived. In particular, the differentiability of a class of tracking-type functionals was shown. We build on these results to derive an adjoint gradient representation for such functionals. The adjoint problem is a system of linear transport equations with a discontinuous coefficient and discontinuous terminal state. We prove the existence and uniqueness of a piecewise Lipschitz continuous solution to the adjoint problem which satisfies suitable interior boundary conditions along all shock curves. The adjoint state generally contains infinitely many discontinuities but its total variation remains uniformly bounded in time. AB - We derive an adjoint gradient representation for the objective functional of an optimal control problem of the unique entropy solution to the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation or balance laws. The GRP is an initial value problem where the initial state is piecewise C^1 with exactly one discontinuity. In a recent work, differentiability properties of the solution operator of the GRP were derived. In particular, the differentiability of a class of tracking-type functionals was shown. We build on these results to derive an adjoint gradient representation for such functionals. The adjoint problem is a system of linear transport equations with a discontinuous coefficient and discontinuous terminal state. We prove the existence and uniqueness of a piecewise Lipschitz continuous solution to the adjoint problem which satisfies suitable interior boundary conditions along all shock curves. The adjoint state generally contains infinitely many discontinuities but its total variation remains uniformly bounded in time. KW - optimal control KW - adjoint calculus KW - hyperbolic systems of conservation laws Y1 - 2026 SP - 50 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Karsai, Attila A1 - Schulze, Philipp T1 - A discrete gradient scheme for preserving QSR-dissipativity N2 - The notion of dissipative dynamical systems provides a formal description of processes that cannot generate energy internally. For these systems, changes in energy can only occur due to an external energy supply or dissipation effects. Unfortunately, dissipative properties tend to deteriorate in numerical computations, especially in nonlinear systems. Discrete gradient methods can help mitigate this problem. In this paper, we present a class of structure-preserving time discretization schemes based on discrete gradients for a special class of systems that are dissipative with respect to a quadratic supply rate. AB - The notion of dissipative dynamical systems provides a formal description of processes that cannot generate energy internally. For these systems, changes in energy can only occur due to an external energy supply or dissipation effects. Unfortunately, dissipative properties tend to deteriorate in numerical computations, especially in nonlinear systems. Discrete gradient methods can help mitigate this problem. In this paper, we present a class of structure-preserving time discretization schemes based on discrete gradients for a special class of systems that are dissipative with respect to a quadratic supply rate. Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2602.15445 DO - https://doi.org/10.48550/arXiv.2602.15445 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Altmann, Robert A1 - Karsai, Attila A1 - Schulze, Philipp T1 - Structure-Preserving Discretization and Model Reduction for Energy-Based Models N2 - We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our framework combines existing ideas from the literature and systematically addresses temporal discretization, spatial discretization, and model order reduction, ensuring that all resulting schemes are dissipation-preserving in the sense of a discrete dissipation inequality. For this, we use a Petrov-Galerkin ansatz together with appropriate projections. Numerical results for a nonlinear circuit model and the Cahn-Hilliard equation illustrate the effectiveness of the approach. AB - We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our framework combines existing ideas from the literature and systematically addresses temporal discretization, spatial discretization, and model order reduction, ensuring that all resulting schemes are dissipation-preserving in the sense of a discrete dissipation inequality. For this, we use a Petrov-Galerkin ansatz together with appropriate projections. Numerical results for a nonlinear circuit model and the Cahn-Hilliard equation illustrate the effectiveness of the approach. Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2507.21552 DO - https://doi.org/10.48550/arXiv.2507.21552 ER - TY - THES U1 - Dissertation oder Habilitation A1 - Karsai, Attila T1 - Nonlinear energy-based systems: modeling, control and numerical realization N2 - Dynamical systems consist of ordinary and partial differential equations and are among the most prominent approaches to model physical processes. They describe the evolution of the system in terms of the current system state and external control inputs. To improve model accuracy, it can be beneficial to explicitly include properties such as energy conservation or energy dissipation, which are usually present in the real-world phenomena, in the mathematical problem description. Ideally, these properties should be kept in mind whenever one interacts with the model. The focus of this thesis is on two kinds of interactions: the discretization of such models, and their control. Discretization techniques are necessary whenever the system evolution is to be approximated using computational methods. Similarly fundamental is the numerical realization of control inputs that lead to desired outcomes. Both aspects require special attention to retain an energy-based perspective. For this, the first step is usually to encode the energy properties in the algebraic description of the model. This description must strike a balance between general applicability and its corresponding benefits. The next step is to leverage the algebraic description in further analysis. While the field is well-developed for linear systems, the nonlinear case often poses additional difficulties. First, there seems to be no clear consensus about what model class to use to describe nonlinear physical phenomena. Second, many discretization methods that preserve the energy-based viewpoint in the linear case are not trivial to generalize to nonlinear systems. Third, studying the behavior of energy-optimal controls and finding control laws that can be realized via energy-based models is usually more difficult. In this thesis, these points are addressed. We give an overview of energy-based model classes used for nonlinear phenomena in both finite and infinite dimensions and investigate their relationship. Moreover, using a modified Petrov--Galerkin method and discrete gradients, we present multiple structure-preserving discretization schemes. Furthermore, we show that similar to the linear case, energy-optimal controls steer the associated trajectories to the submanifold of the state space where no dissipation is present. Finally, we combine an optimal feedback law characterized by the Hamilton--Jacobi--Bellman equation with output feedback to state an energy-based feedback controller. Our theoretical results are illustrated using numerical experiments. AB - Dynamical systems consist of ordinary and partial differential equations and are among the most prominent approaches to model physical processes. They describe the evolution of the system in terms of the current system state and external control inputs. To improve model accuracy, it can be beneficial to explicitly include properties such as energy conservation or energy dissipation, which are usually present in the real-world phenomena, in the mathematical problem description. Ideally, these properties should be kept in mind whenever one interacts with the model. The focus of this thesis is on two kinds of interactions: the discretization of such models, and their control. Discretization techniques are necessary whenever the system evolution is to be approximated using computational methods. Similarly fundamental is the numerical realization of control inputs that lead to desired outcomes. Both aspects require special attention to retain an energy-based perspective. For this, the first step is usually to encode the energy properties in the algebraic description of the model. This description must strike a balance between general applicability and its corresponding benefits. The next step is to leverage the algebraic description in further analysis. While the field is well-developed for linear systems, the nonlinear case often poses additional difficulties. First, there seems to be no clear consensus about what model class to use to describe nonlinear physical phenomena. Second, many discretization methods that preserve the energy-based viewpoint in the linear case are not trivial to generalize to nonlinear systems. Third, studying the behavior of energy-optimal controls and finding control laws that can be realized via energy-based models is usually more difficult. In this thesis, these points are addressed. We give an overview of energy-based model classes used for nonlinear phenomena in both finite and infinite dimensions and investigate their relationship. Moreover, using a modified Petrov--Galerkin method and discrete gradients, we present multiple structure-preserving discretization schemes. Furthermore, we show that similar to the linear case, energy-optimal controls steer the associated trajectories to the submanifold of the state space where no dissipation is present. Finally, we combine an optimal feedback law characterized by the Hamilton--Jacobi--Bellman equation with output feedback to state an energy-based feedback controller. Our theoretical results are illustrated using numerical experiments. Y1 - 2026 U6 - https://doi.org/10.14279/depositonce-25690 DO - https://doi.org/10.14279/depositonce-25690 ER - TY - INPR U1 - Preprint A1 - Geiselmann, Zoe A1 - Joswig, Michael A1 - Kastner, Lars A1 - Mundinger, Konrad A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Wack, Marcel A1 - Zimmer, Max T1 - 121 Patchworked Curves of Degree Seven N2 - The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle 7⋅Δ2 for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996). AB - The 121 real schemes, i.e., ambient isotopy classes, of smooth real plane algebraic curves of degree seven were classified by Viro (1984). By constructing one patchwork of the dilated triangle 7⋅Δ2 for each real scheme, we provide an explicit method for constructing polynomials realizing each real scheme. In particular, every real scheme of degree seven can be realized as a T-curve; this settles a question raised by Itenberg and Viro (1996). Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Geiselmann, Zoe A1 - Joswig, Michael A1 - Kastner, Lars A1 - Mundinger, Konrad A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Wack, Marcel A1 - Zimmer, Max T1 - Fast Isotopy Computation for T-Curves N2 - A T-curve of degree d is given by a regular unimodular triangulation of d⋅Δ2 together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves. AB - A T-curve of degree d is given by a regular unimodular triangulation of d⋅Δ2 together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves. Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Göß, Adrian T1 - Clash of MINLP Relaxations: Piecewise Linear vs. Global Parabolic N2 - Solving mixed-integer nonlinear programs (MINLPs) typically relies on constructing relaxations that are easier to tackle than the original problem. Recently, global parabolic (PARA) relaxations were introduced, featuring separable quadratic functions – paraboloids – as global under- or overestimators of general nonlinear constraint functions. So far, the paraboloids are all computed at once by solving a mixed-integer linear program (MIP). For small tolerances or wide function domains, the corresponding MIP grows in size and is eventually intractable, preventing a meaningful comparison with established relaxation techniques. We therefore propose a novel iterative method to compute PARA approximations that succeeds on all tolerance-domain combinations where the original one has failed. The computational study is preceded by a thorough theoretical explanation and analysis. Finally, the improved method enables a computational comparison with piecewise linear (PWL) relaxations in terms of runtime on general MINLP instances. The results show that the modern solver SCIP can solve PWL relaxations faster when the olerance is high, shifting strongly in favor of PARA for tighter tolerances. We attribute the effect to the difference in the corresponding problem size: PWL relaxations introduce binary variables to identify the active linear piece and their number grows with decreasing tolerance. PARA, on the other hand, does not require additional variables such that the dimension is maintained. For problems with at least one (co)sine constraint, the effect significantly amplifies. Thereby, for medium tolerances, PARA relaxations outperform SCIP stand-alone. Applied problems like alternating current optimal power flow (AC-OPF) feature such constraint types, leaving PARA a viable relaxation strategy. AB - Solving mixed-integer nonlinear programs (MINLPs) typically relies on constructing relaxations that are easier to tackle than the original problem. Recently, global parabolic (PARA) relaxations were introduced, featuring separable quadratic functions – paraboloids – as global under- or overestimators of general nonlinear constraint functions. So far, the paraboloids are all computed at once by solving a mixed-integer linear program (MIP). For small tolerances or wide function domains, the corresponding MIP grows in size and is eventually intractable, preventing a meaningful comparison with established relaxation techniques. We therefore propose a novel iterative method to compute PARA approximations that succeeds on all tolerance-domain combinations where the original one has failed. The computational study is preceded by a thorough theoretical explanation and analysis. Finally, the improved method enables a computational comparison with piecewise linear (PWL) relaxations in terms of runtime on general MINLP instances. The results show that the modern solver SCIP can solve PWL relaxations faster when the olerance is high, shifting strongly in favor of PARA for tighter tolerances. We attribute the effect to the difference in the corresponding problem size: PWL relaxations introduce binary variables to identify the active linear piece and their number grows with decreasing tolerance. PARA, on the other hand, does not require additional variables such that the dimension is maintained. For problems with at least one (co)sine constraint, the effect significantly amplifies. Thereby, for medium tolerances, PARA relaxations outperform SCIP stand-alone. Applied problems like alternating current optimal power flow (AC-OPF) feature such constraint types, leaving PARA a viable relaxation strategy. Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Klimm, Max A1 - Pfetsch, Marc E. A1 - Skutella, Martin A1 - Strubberg, Lea T1 - Approximating the Network Design Problem for Potential-Based Flows N2 - We develop efficient algorithms for a fundamental network design problem arising in potential-based flow models, which are central to many energy transport networks (e.g., hydrogen and electricity). In contrast to classical network flow problems, the nonlinearities inherent in potential-based networks introduce significant new challenges. We address these challenges through intricate reductions to classical combinatorial optimization problems, such as (constrained) shortest path problems, enabling the application of well-established algorithmic techniques to compute exact and approximate solutions efficiently. Finally, we complement these algorithmic results with matching complexity results concerning the hardness and non-approximability of the considered problem variants. AB - We develop efficient algorithms for a fundamental network design problem arising in potential-based flow models, which are central to many energy transport networks (e.g., hydrogen and electricity). In contrast to classical network flow problems, the nonlinearities inherent in potential-based networks introduce significant new challenges. We address these challenges through intricate reductions to classical combinatorial optimization problems, such as (constrained) shortest path problems, enabling the application of well-established algorithmic techniques to compute exact and approximate solutions efficiently. Finally, we complement these algorithmic results with matching complexity results concerning the hardness and non-approximability of the considered problem variants. Y1 - 2026 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Zimmer, Max A1 - Mundinger, Konrad T1 - Extending the Continuum of Six-Colorings JF - Geocombinatorics Quarterly N2 - We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach. AB - We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach. Y1 - 2024 U6 - https://doi.org/10.48550 DO - https://doi.org/10.48550 VL - Geocombinatorics Quarterly IS - Volume XXXIV: 2024 SP - 10 ER - TY - CPAPER U1 - Konferenzveröffentlichung A1 - Pokutta, Sebastian A1 - Zimmer, Max A1 - Mundinger, Konrad T1 - Neural Parameter Regression for Explicit Representations of PDE Solution Operators N2 - We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et. al, 2021) by employing Physics-Informed Neural Network (Raissi et. al, 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples. AB - We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et. al, 2021) by employing Physics-Informed Neural Network (Raissi et. al, 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples. Y1 - 2024 SP - 15 ER - TY - CPAPER U1 - Konferenzveröffentlichung A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Zimmer, Max A1 - Kiem, Aldo A1 - Mundinger, Konrad T1 - Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? N2 - We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights. AB - We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights. Y1 - 2025 SP - 20 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Chaumet, Aidan A1 - Giesselmann, Jan T1 - Convergence Analysis of a Fully Discrete Observer for Data Assimilation of the Barotropic Euler Equations N2 - We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations. AB - We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations. KW - Data Assimilation KW - Observer KW - Relative Energy KW - Euler Equations KW - Fully Discrete Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2603.10962 DO - https://doi.org/10.48550/arXiv.2603.10962 ER - TY - JFULL U1 - Periodikum A1 - Börner, Pascal A1 - Pfetsch, Marc E. A1 - Ulbrich, Stefan T1 - Mixing of Gases in Stationary Networks: Properties and Optimization N2 - This paper deals with the mixing of gases in stationary networks. We first derive a model and pressure law for mixing, for which there is empirical evidence for its accuracy. The model is based on the change of the speed of sound in gas mixtures. We then consider stationary gas networks. The existence result for solutions of single gas flow on networks is extended to mixtures. Further, we establish an easy to check criterion for uniqueness of solutions on the network. This results in a uniqueness proof of solutions on all networks with a mixture of natural gas and low hydrogen percentages. We then develop a solution algorithm that alternates between the solution of a single gas problem and update of the mixture ratios. In a computational study, different model variants and their impact on performance are compared. Moreover, the increased complexity of solving stationary gas transport problems with mixing is evaluated. AB - This paper deals with the mixing of gases in stationary networks. We first derive a model and pressure law for mixing, for which there is empirical evidence for its accuracy. The model is based on the change of the speed of sound in gas mixtures. We then consider stationary gas networks. The existence result for solutions of single gas flow on networks is extended to mixtures. Further, we establish an easy to check criterion for uniqueness of solutions on the network. This results in a uniqueness proof of solutions on all networks with a mixture of natural gas and low hydrogen percentages. We then develop a solution algorithm that alternates between the solution of a single gas problem and update of the mixture ratios. In a computational study, different model variants and their impact on performance are compared. Moreover, the increased complexity of solving stationary gas transport problems with mixing is evaluated. KW - gas network optimization KW - gas mixing KW - MINLP KW - global optimization Y1 - 2026 SP - 44 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Henrion, René A1 - Schmidt, Martin T1 - Chance-Constrained Linear Complementarity Problems N2 - We study linear complementarity problems (LCPs) under uncertainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relaxations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove convexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small example of a Cournot oligopoly among energy producers, we finally illustrate our theoretical findings. AB - We study linear complementarity problems (LCPs) under uncertainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relaxations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove convexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small example of a Cournot oligopoly among energy producers, we finally illustrate our theoretical findings. KW - Linear complementarity problems KW - Chance constraints KW - Existence KW - Convexity KW - Optimality conditions Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Breiten, Tobias A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Hiller, Benjamin A1 - Karsai, Attila A1 - Lang, Jens A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Tischendorf, Caren A1 - Tscherpel, Tabea T1 - A Catalog of Gas Network Models: PDEs, Coupling Conditions, and Numerical Schemes N2 - This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems. We do not claim completeness and refer in many places to the existing literature. AB - This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems. We do not claim completeness and refer in many places to the existing literature. Y1 - 2026 N1 - This is an updated version of [P. Domschke, B. Hiller, J. Lang, V. Mehrmann, R. Morandin, and C. Tischendorf. Gas Network Modeling: An Overview. Preprint, TRR 154, 2021], available at: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/411 ER - TY - INPR U1 - Preprint A1 - Börner, Pascal A1 - Giesselmann, Jan A1 - Kumar, Varun M. A1 - Pfetsch, Marc E. A1 - Thiele, Michael A1 - Tscherpel, Tabea T1 - Gas Mixtures on Networks: Modeling, Simulation and Optimization N2 - This chapter addresses mathematical models for isothermal mixtures of hydrogen and natural gas, motivated by the need for reliable simulation tools in future low-carbon energy systems. We analyze several classes of mixture models and investigate their convergence properties in the regime of strong interaction between constituents, covering stationary and instationary single-pipe settings as well as network flows. Since mixture models critically depend on the choice of pressure law, we compare the industry-standard GERG equation of state with simplified alternatives that preserve convex energies and reduce computational costs. For network applications, we discuss consistent coupling conditions across model classes, explore optimization of steady flows using the algebraic Weymouth formulation, and provide numerical evidence for its applicability in relevant operating regimes. The study reveals when simplified models are justified and outlines key open challenges for the modeling of gas mixtures. AB - This chapter addresses mathematical models for isothermal mixtures of hydrogen and natural gas, motivated by the need for reliable simulation tools in future low-carbon energy systems. We analyze several classes of mixture models and investigate their convergence properties in the regime of strong interaction between constituents, covering stationary and instationary single-pipe settings as well as network flows. Since mixture models critically depend on the choice of pressure law, we compare the industry-standard GERG equation of state with simplified alternatives that preserve convex energies and reduce computational costs. For network applications, we discuss consistent coupling conditions across model classes, explore optimization of steady flows using the algebraic Weymouth formulation, and provide numerical evidence for its applicability in relevant operating regimes. The study reveals when simplified models are justified and outlines key open challenges for the modeling of gas mixtures. Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Brunk, Aaron A1 - Giesselmann, Jan A1 - Tscherpel, Tabea T1 - A posteriori existence of strong solutions to the Navier-Stokes equations in 3D N2 - Global existence of strong solutions to the three-dimensional incompressible Navier--Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out blow-up on a certain time interval, using only numerical solutions. In this work we present such a result for the Navier--Stokes equations subject to periodic boundary conditions, which makes use of a version of the celebrated blow-up criterion in the critical space $L^\infty(L^3)$ by Iskauriaza, Serëgin and Shverak (2003). Our approach is based on a conditional stability estimate in $L^2$ and $L^3$. The a posteriori criterion that, if satisfied, verifies existence of strong solutions, involves only negative Sobolev norms of the residual. We apply the criterion to numerical approximations computed with mixed finite elements and an implicit Euler time discretisation. A posteriori error estimates allow us to derive a fully computable criterion without imposing any extra assumptions on the solution. While limited to short time intervals, with sufficient computational resources in principle the criterion might allow for a verification over longer time intervals than what can be achieved by theoretical means. AB - Global existence of strong solutions to the three-dimensional incompressible Navier--Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out blow-up on a certain time interval, using only numerical solutions. In this work we present such a result for the Navier--Stokes equations subject to periodic boundary conditions, which makes use of a version of the celebrated blow-up criterion in the critical space $L^\infty(L^3)$ by Iskauriaza, Serëgin and Shverak (2003). Our approach is based on a conditional stability estimate in $L^2$ and $L^3$. The a posteriori criterion that, if satisfied, verifies existence of strong solutions, involves only negative Sobolev norms of the residual. We apply the criterion to numerical approximations computed with mixed finite elements and an implicit Euler time discretisation. A posteriori error estimates allow us to derive a fully computable criterion without imposing any extra assumptions on the solution. While limited to short time intervals, with sufficient computational resources in principle the criterion might allow for a verification over longer time intervals than what can be achieved by theoretical means. KW - Navier-Stokes KW - blow-up KW - a posteriori estimates KW - critical space KW - reconstruction Y1 - 2026 ER - TY - INPR U1 - Preprint A1 - Göttlich, Simone A1 - Schuster, Michael A1 - Ulke, Alena T1 - On the Existence of Steady States for Blended Gas Flow with Non-Constant Compressibility Factor on Networks N2 - In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states. AB - In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states. KW - Blended Gas Flow KW - Compressibility Factor KW - z-Factor KW - Real Gas KW - Steady States Y1 - 2026 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Heitsch, Holger A1 - Henrion, René T1 - On the Lipschitz continuity of the spherical cap discrepancy around generic point sets JF - Unif. Distrib. Theory N2 - The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy. AB - The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - Lipschitz continuity KW - necessary optimality conditions Y1 - 2026 U6 - https://doi.org/10.2478/udt-2025-0011 DO - https://doi.org/10.2478/udt-2025-0011 VL - 20 IS - 1 SP - 35 EP - 63 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Bernhard, Daniela A1 - Heitsch, Holger A1 - Henrion, René A1 - Liers, Frauke A1 - Stingl, Michael A1 - Uihlein, Andrian A1 - Zipf, Viktor T1 - Continuous stochastic gradient and spherical radial decomposition N2 - In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks. AB - In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks. KW - chance constraints KW - continuous stochastic gradient KW - spheric-radial decomposition Y1 - 2026 ER - TY - THES U1 - Habilitation A1 - Schuster, Michael T1 - Control and Optimization under Uncertainty in the Context of Gas Network Operation N2 - The transition to renewable energy and the increasing role of hydrogen as a future energy carrier pose major challenges for the operation and optimization of gas transport networks. A rigorous mathematical understanding of the topic is necessary as efficient and reliable operation requires advanced methods to deal with uncertainty, nonlinear dynamics, and complex network topologies. At the same time, optimal control theory – in particular the turnpike phenomenon – provides powerful tools to simplify long-term optimization problems and to connect dynamic models with their stationary counterparts. This habilitation thesis focuses on establishing new results in three related areas. For optimization problems with probabilistic constraints, an approach for approximating probabilities based on kernel density estimation is presented and analyzed, yielding necessary and sufficient conditions for the convergence of solutions of approximated stochastic optimization problems. In the modeling of gas transport networks, the existence and uniqueness results of a mixing model are presented. Furthermore, a finite-time turnpike result for an optimal control problem governed by the wave equation as well as an integral turnpike result for an optimal control problem governed by the transport equation under uncertainty is established. On the application side, the problem of optimal placement of compressor stations in stationary gas networks is analyzed for both deterministic and random gas demand. Results on the optimal number of compressor stations and their locations on gas networks are presented. Further, the turnpike phenomenon is applied to an optimal control and design problem for gas networks, allowing steady-state models to replace the gas dynamics in the long-term planning. For the corresponding stationary problem, a fast and efficient algorithm for identifying the optimal network topology with low control cost is provided. AB - The transition to renewable energy and the increasing role of hydrogen as a future energy carrier pose major challenges for the operation and optimization of gas transport networks. A rigorous mathematical understanding of the topic is necessary as efficient and reliable operation requires advanced methods to deal with uncertainty, nonlinear dynamics, and complex network topologies. At the same time, optimal control theory – in particular the turnpike phenomenon – provides powerful tools to simplify long-term optimization problems and to connect dynamic models with their stationary counterparts. This habilitation thesis focuses on establishing new results in three related areas. For optimization problems with probabilistic constraints, an approach for approximating probabilities based on kernel density estimation is presented and analyzed, yielding necessary and sufficient conditions for the convergence of solutions of approximated stochastic optimization problems. In the modeling of gas transport networks, the existence and uniqueness results of a mixing model are presented. Furthermore, a finite-time turnpike result for an optimal control problem governed by the wave equation as well as an integral turnpike result for an optimal control problem governed by the transport equation under uncertainty is established. On the application side, the problem of optimal placement of compressor stations in stationary gas networks is analyzed for both deterministic and random gas demand. Results on the optimal number of compressor stations and their locations on gas networks are presented. Further, the turnpike phenomenon is applied to an optimal control and design problem for gas networks, allowing steady-state models to replace the gas dynamics in the long-term planning. For the corresponding stationary problem, a fast and efficient algorithm for identifying the optimal network topology with low control cost is provided. N2 - Die Energiewende und die zunehmende Bedeutung von Wasserstoff als zukünftiger Energieträger stellen erhebliche Herausforderungen an den Betrieb und die Optimierung von Gastransportnetzen. Ein detailliertes, mathematisches Verständnis ist notwendig, da ein effizienter und sicherer Betrieb komplexe Methoden im Umgang mit Unsicherheiten, nichtlinearen Transportdynamiken und komplexen Netzstrukturen erfordert. Gleichzeitig liefert die Optimalsteuerungstheorie – insbesondere das Turnpike-Phänomen – wirkungsvolle Werkzeuge, um langzeitorientierte Optimierungsprobleme zu vereinfachen und dynamische Modelle mit ihren stationären Gegenstücken zu verbinden. Diese Habilitationsschrift konzentriert sich auf die Entwicklung neuer Ergebnisse in drei verwandten Forschungsbereichen. Für Optimierungsprobleme mit probabilistischen Nebenbedingungen wird ein Ansatz zur Approximation von Wahrscheinlichkeiten basierend auf Kerndichteschätzern präsentiert und analysiert, der notwendige und hinreichende Bedingungen für die Konvergenz von Lösungen approximierter stochastischer Optimierungsprobleme liefert. In der Modellierung von Gastransportnetzen werden Existenz- und Eindeutigkeitsresultate für ein Mischungsmodell hergeleitet. Darüber hinaus wird ein Finite-Time-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit der Wellengleichung sowie ein Integral-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit Transportgleichungen unter Unsicherheit präsentiert. Auf der Anwendungsseite wird das Problem der optimalen Platzierung von Verdichterstationen in stationären Gasnetzen sowohl für deterministische als auch für stochastische Gasnachfrage untersucht. Ergebnisse zur optimalen Anzahl und Positionierung von Verdichtern werden vorgestellt. Ferner wird das Turnpike-Phänomen auf ein kombiniertes Steuerungs- und Designproblem für Gasnetze angewandt, wodurch stationäre Modelle anstelle der Gasdynamik in der langfristigen Planung genutzt werden können. Für das entsprechende stationäre Problem wird ein schnelles und effizientes Verfahren zur Bestimmung optimaler Netzwerktopologien mit geringen Steuerungskosten bereitgestellt. AB - Die Energiewende und die zunehmende Bedeutung von Wasserstoff als zukünftiger Energieträger stellen erhebliche Herausforderungen an den Betrieb und die Optimierung von Gastransportnetzen. Ein detailliertes, mathematisches Verständnis ist notwendig, da ein effizienter und sicherer Betrieb komplexe Methoden im Umgang mit Unsicherheiten, nichtlinearen Transportdynamiken und komplexen Netzstrukturen erfordert. Gleichzeitig liefert die Optimalsteuerungstheorie – insbesondere das Turnpike-Phänomen – wirkungsvolle Werkzeuge, um langzeitorientierte Optimierungsprobleme zu vereinfachen und dynamische Modelle mit ihren stationären Gegenstücken zu verbinden. Diese Habilitationsschrift konzentriert sich auf die Entwicklung neuer Ergebnisse in drei verwandten Forschungsbereichen. Für Optimierungsprobleme mit probabilistischen Nebenbedingungen wird ein Ansatz zur Approximation von Wahrscheinlichkeiten basierend auf Kerndichteschätzern präsentiert und analysiert, der notwendige und hinreichende Bedingungen für die Konvergenz von Lösungen approximierter stochastischer Optimierungsprobleme liefert. In der Modellierung von Gastransportnetzen werden Existenz- und Eindeutigkeitsresultate für ein Mischungsmodell hergeleitet. Darüber hinaus wird ein Finite-Time-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit der Wellengleichung sowie ein Integral-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit Transportgleichungen unter Unsicherheit präsentiert. Auf der Anwendungsseite wird das Problem der optimalen Platzierung von Verdichterstationen in stationären Gasnetzen sowohl für deterministische als auch für stochastische Gasnachfrage untersucht. Ergebnisse zur optimalen Anzahl und Positionierung von Verdichtern werden vorgestellt. Ferner wird das Turnpike-Phänomen auf ein kombiniertes Steuerungs- und Designproblem für Gasnetze angewandt, wodurch stationäre Modelle anstelle der Gasdynamik in der langfristigen Planung genutzt werden können. Für das entsprechende stationäre Problem wird ein schnelles und effizientes Verfahren zur Bestimmung optimaler Netzwerktopologien mit geringen Steuerungskosten bereitgestellt. Y1 - 2026 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin T1 - Synchronization of velocities in pipeline flow of blended gas JF - Journal of Mathematical Analysis and Applications N2 - We consider the pipeline flflow of blended gas. The flow is governed by a coupled system where for each component we have the isothermal Euler equations with an additional velocity coupling term that couples the velocities of the different components. Our motivation is hydrogen blending in natural gas pipelines, which will play a role in the transition to renewable energies. We show that with suitable boundary conditions the velocities of the gas components synchronize exponentially fast, as long as the L2-norm of the synchronization error is outside of a certain interval where the size of the interval is determined by the order of the interaction terms. This indicates that in some cases for a mixture of ncomponents it is justifified to use a flux model where it is assumed that all components flow with the same velocity. For the proofs we use an appropriately chosen Lyapunov function which is based upon the idea of relative energy. AB - We consider the pipeline flflow of blended gas. The flow is governed by a coupled system where for each component we have the isothermal Euler equations with an additional velocity coupling term that couples the velocities of the different components. Our motivation is hydrogen blending in natural gas pipelines, which will play a role in the transition to renewable energies. We show that with suitable boundary conditions the velocities of the gas components synchronize exponentially fast, as long as the L2-norm of the synchronization error is outside of a certain interval where the size of the interval is determined by the order of the interaction terms. This indicates that in some cases for a mixture of ncomponents it is justifified to use a flux model where it is assumed that all components flow with the same velocity. For the proofs we use an appropriately chosen Lyapunov function which is based upon the idea of relative energy. KW - Synchronization of solutions to PDEs KW - Quasi-linear hyperbolic PDE KW - Drift-flux model KW - Lyapunov function KW - Relative energy Y1 - 2026 U6 - https://doi.org/10.1016/j.jmaa.2025.130078 DO - https://doi.org/10.1016/j.jmaa.2025.130078 VL - 556 IS - 1 SP - 19 ER - TY - CHAP U1 - Teil eines Buches A1 - Pfetsch, Marc A1 - Schmidt, Martin A1 - Skutella, Martin A1 - Thürauf, Johannes T1 - Potential-Based Flows - An Overview T2 - Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks N2 - Potential-based flows provide an algebraic way to model static physical flows in networks, for example, in gas, water, and lossless DC power networks. The flow on an arc in the network depends on the difference of the potentials at its end-nodes, possibly in a nonlinear way. Potential-based flows have several nice properties like uniqueness and acyclicity. The goal of this paper is to provide an overview of the current knowledge on these models with a focus on optimization problems on such networks. We cover basic properties, computational complexity, monotonicity, uncertain parameters, and the corresponding behavior of the network as well as topology optimization. AB - Potential-based flows provide an algebraic way to model static physical flows in networks, for example, in gas, water, and lossless DC power networks. The flow on an arc in the network depends on the difference of the potentials at its end-nodes, possibly in a nonlinear way. Potential-based flows have several nice properties like uniqueness and acyclicity. The goal of this paper is to provide an overview of the current knowledge on these models with a focus on optimization problems on such networks. We cover basic properties, computational complexity, monotonicity, uncertain parameters, and the corresponding behavior of the network as well as topology optimization. Y1 - 2026 PB - Springer-Nature ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin T1 - Boundary Stabilization of Quasi-Linear Hyperbolic Systems with Varying Time-Delay N2 - In this paper we consider the boundary feedback stabilization of a quasi-linear hyperbolic system of balance laws. At one end of the space interval, there is a reflecting boundary condition. At the other end a stabilizing feedback law with a varying time-delay is prescribed. We present sufficient conditions for the exponential stability of the system. We show that exponential stabilization is possible if the product of the length of the interval and an upper bound for the source term is sufficiently small. We also show that if the product of the length of the interval and a lower bound for the source term is sufficiently large, the system is unstable. Our analysis is based on Lyapunov functions with weights that are given by hyperbolic functions that generalize the well-known exponential weights. Compared with previous contributions, we obtain conditions that can be verified more easily in terms of the system parameters. Our results show that for sufficiently short space intervals, and also with varying time-delay, exponential stabilization is possible with appropriately chosen feedback gains that depend on the maximal value of the time-delay and the maximal absolute value of its derivative. AB - In this paper we consider the boundary feedback stabilization of a quasi-linear hyperbolic system of balance laws. At one end of the space interval, there is a reflecting boundary condition. At the other end a stabilizing feedback law with a varying time-delay is prescribed. We present sufficient conditions for the exponential stability of the system. We show that exponential stabilization is possible if the product of the length of the interval and an upper bound for the source term is sufficiently small. We also show that if the product of the length of the interval and a lower bound for the source term is sufficiently large, the system is unstable. Our analysis is based on Lyapunov functions with weights that are given by hyperbolic functions that generalize the well-known exponential weights. Compared with previous contributions, we obtain conditions that can be verified more easily in terms of the system parameters. Our results show that for sufficiently short space intervals, and also with varying time-delay, exponential stabilization is possible with appropriately chosen feedback gains that depend on the maximal value of the time-delay and the maximal absolute value of its derivative. KW - boundary stabilization KW - feedback law KW - varying delay KW - quasi-linear hyperbolic system KW - boundary control Y1 - 2025 U6 - https://doi.org/10.1137/24M1648570 DO - https://doi.org/10.1137/24M1648570 VL - 63 IS - SIAM Journal on Control and Optimization SP - 452 EP - 471 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Ranocha, Hendrik T1 - Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions N2 - We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings. AB - We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings. Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Breitkopf, Jannik A1 - Gugat, Martin A1 - Ulbrich, Stefan T1 - Existence and Optimal Boundary Control of Classical Solutions to Networks of Quasilinear Hyperbolic Systems of Balance Laws N2 - We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels. It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones. We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls. AB - We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels. It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones. We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls. KW - networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control Y1 - 2025 SP - 42 ER - TY - INPR U1 - Preprint A1 - Bernhard, Daniela A1 - Liers, Frauke A1 - Stingl, Michael T1 - Branch-and-cut for mixed-integer robust chance-constrained optimization with discrete distributions N2 - We study robust chance-constrained problems with mixed-integer design variables and ambiguity sets consisting of discrete probability distributions. Allowing general non-convex constraint functions, we develop a branch-and-cut framework using scenario-based cutting planes to generate lower bounds. The cutting planes are obtained by exploiting the classical big-M reformulation of the chance-constrained problem in the case of discrete distributions. Furthermore, we include the calculation of initial feasible solutions based on a bundle method applied to an approximation of the original problem into the branch-and-cut procedure. We conclude with a detailed discussion about the practical performance of the branch-and-cut framework with and without initial feasible solutions. In our experiments we focus on gas transport problems under uncertainty and provide a comparison of our method with solving the classical reformulation directly for various real-world sized instances. AB - We study robust chance-constrained problems with mixed-integer design variables and ambiguity sets consisting of discrete probability distributions. Allowing general non-convex constraint functions, we develop a branch-and-cut framework using scenario-based cutting planes to generate lower bounds. The cutting planes are obtained by exploiting the classical big-M reformulation of the chance-constrained problem in the case of discrete distributions. Furthermore, we include the calculation of initial feasible solutions based on a bundle method applied to an approximation of the original problem into the branch-and-cut procedure. We conclude with a detailed discussion about the practical performance of the branch-and-cut framework with and without initial feasible solutions. In our experiments we focus on gas transport problems under uncertainty and provide a comparison of our method with solving the classical reformulation directly for various real-world sized instances. Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Breitkopf, Jannik A1 - Ulbrich, Stefan T1 - A Variational Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws N2 - We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable (C^1) states left and right of it. The control consists of the C^1 parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon T>0 to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small. Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to C^0. In physical coordinates, this implies that the shock curves in C^1 and the states between the shocks in the topology of C^0 depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals. AB - We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable (C^1) states left and right of it. The control consists of the C^1 parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon T>0 to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small. Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to C^0. In physical coordinates, this implies that the shock curves in C^1 and the states between the shocks in the topology of C^0 depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals. KW - hyperbolic systems of conservation laws, shock curves, generalized riemann problem, optimal control, variational calculus Y1 - 2025 SP - 34 ER - TY - INPR U1 - Preprint A1 - Denzler, Sebastian A1 - Aigner, Kevin-Martin A1 - Lüer, Larry A1 - Brabec, Christoph A1 - Liers, Frauke T1 - Robust Bayesian Optimization with an Application to Material Science N2 - We propose a novel online learning framework for robust Bayesian optimization of uncertain black-box functions. While Bayesian optimization is well-suited for data-efficient optimization of expensive objectives, its standard form can be sensitive to hidden or varying parameters. To address this issue, we consider a min–max robust counterpart of the optimization problem and develop a practically efficient solution algorithm, BROVER (Bayesian Robust Optimization via Exploration with Regret minimization). Our method combines Gaussian process regression with a decomposition approach: the minimax structure is split into a non-convex online learner based on the Follow-the-Perturbed-Leader algorithm together with a subsequent minimization step in the decision variables. We prove that the theoretical regret bound converges under mild assumptions, ensuring asymptotic convergence to robust solutions. Numerical experiments on synthetic data validate the regret guarantees and demonstrate fast convergence to the robust optimum. Furthermore, we apply our method to the robust optimization of organic solar cell performance, where hidden process parameters and experimental variability naturally induce uncertainty. Our results on real-world datae show that BROVER identifies solutions with strong robustness properties within relatively few iterations, thereby offering a modern and practical approach for data-driven black-box optimization under uncertainty. AB - We propose a novel online learning framework for robust Bayesian optimization of uncertain black-box functions. While Bayesian optimization is well-suited for data-efficient optimization of expensive objectives, its standard form can be sensitive to hidden or varying parameters. To address this issue, we consider a min–max robust counterpart of the optimization problem and develop a practically efficient solution algorithm, BROVER (Bayesian Robust Optimization via Exploration with Regret minimization). Our method combines Gaussian process regression with a decomposition approach: the minimax structure is split into a non-convex online learner based on the Follow-the-Perturbed-Leader algorithm together with a subsequent minimization step in the decision variables. We prove that the theoretical regret bound converges under mild assumptions, ensuring asymptotic convergence to robust solutions. Numerical experiments on synthetic data validate the regret guarantees and demonstrate fast convergence to the robust optimum. Furthermore, we apply our method to the robust optimization of organic solar cell performance, where hidden process parameters and experimental variability naturally induce uncertainty. Our results on real-world datae show that BROVER identifies solutions with strong robustness properties within relatively few iterations, thereby offering a modern and practical approach for data-driven black-box optimization under uncertainty. KW - robust optimization KW - Bayesian optimization KW - online learning KW - solar cell performance Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Ulke, Alena A1 - Schuster, Michael A1 - Göttlich, Simone T1 - Steady State Blended Gas Flow on Networks: Existence and Uniqueness of Solutions N2 - We prove an existence result for the steady state flow of gas mixtures on networks. The basis of the model are the physical principles of the isothermal Euler equation, coupling conditions for the flow and pressure, and the mixing of incoming flow at nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical properties of the model allowing us to prove the existence of solutions in particular for tree-shaped networks and networks with exactly one cycle. Numerical examples illustrate the results and explore the applicability of our approach to different network topologies. AB - We prove an existence result for the steady state flow of gas mixtures on networks. The basis of the model are the physical principles of the isothermal Euler equation, coupling conditions for the flow and pressure, and the mixing of incoming flow at nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical properties of the model allowing us to prove the existence of solutions in particular for tree-shaped networks and networks with exactly one cycle. Numerical examples illustrate the results and explore the applicability of our approach to different network topologies. KW - gas pipeline network KW - gas mixture KW - hydrogen blending KW - isothermal Euler equations KW - stationary states Y1 - 2024 VL - Netw. Heterog. Media ER - TY - INPR U1 - Preprint A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - The Topological Derivative Method for Optimum Shape Design and Control of Gas Networks N2 - In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations. AB - In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations. KW - Gas Networks KW - Optimum Design KW - Optimal Control KW - Topological Derivative KW - Turnpike Phenomenon Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Wilka, Hendrik A1 - Lang, Jens A1 - Gugat, Martin T1 - Probabilistic Robustness for Compressor Controls in Transient Pipeline Networks N2 - Uncertainty plays a crucial role in modeling and optimization of complex systems across various applications. In this paper, uncertain gas transport through pipeline networks is considered and a novel strategy to measure the robustness of deterministically computed compressor and valve controls, the probabilistic robustness, is presented. \noindent Initially, an optimal control for a deterministic gas network problem is computed such that the total control cost is minimized with respect to box constraints for the pressure. Subsequently, the model is perturbed by uncertain gas demands. The probability, that the uncertain gas pressures - based on the a priori deterministic optimal control - satisfy the box constraints, is evaluated. Moreover, buffer zones are introduced in order to tighten the pressure bounds in the deterministic scenario. Optimal controls for the deterministic scenario with buffer zones are also applied to the uncertain scenario, allowing to analyze the impact of the buffer zones on the probabilistic robustness of the optimal controls. \noindent For the computation of the probability, we apply a kernel density estimator based on samples of the uncertain pressure at chosen locations. In order to reduce the computational effort of generating the samples, we combine the kernel density estimator approach with a stochastic collocation method which approximates the pressure at the chosen locations in the stochastic space. Finally, we discuss generalizations of the probabilistic robustness check and we present numerical results for a gas network taken from the public gas library. AB - Uncertainty plays a crucial role in modeling and optimization of complex systems across various applications. In this paper, uncertain gas transport through pipeline networks is considered and a novel strategy to measure the robustness of deterministically computed compressor and valve controls, the probabilistic robustness, is presented. \noindent Initially, an optimal control for a deterministic gas network problem is computed such that the total control cost is minimized with respect to box constraints for the pressure. Subsequently, the model is perturbed by uncertain gas demands. The probability, that the uncertain gas pressures - based on the a priori deterministic optimal control - satisfy the box constraints, is evaluated. Moreover, buffer zones are introduced in order to tighten the pressure bounds in the deterministic scenario. Optimal controls for the deterministic scenario with buffer zones are also applied to the uncertain scenario, allowing to analyze the impact of the buffer zones on the probabilistic robustness of the optimal controls. \noindent For the computation of the probability, we apply a kernel density estimator based on samples of the uncertain pressure at chosen locations. In order to reduce the computational effort of generating the samples, we combine the kernel density estimator approach with a stochastic collocation method which approximates the pressure at the chosen locations in the stochastic space. Finally, we discuss generalizations of the probabilistic robustness check and we present numerical results for a gas network taken from the public gas library. KW - Probabilistic Robustness KW - Gas Network Control KW - Probabilistic Constrained Optimization KW - Stochastic Collocation KW - Kernel Density Estimation Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Bernhard, Daniela A1 - Stingl, Michael A1 - Liers, Frauke T1 - Algorithms for robust chance-constrained optimization with mixture ambiguity N2 - Constructing ambiguity sets in distributionally robust optimization is difficult and currently receives increased attention. In this paper, we focus on mixture models with finitely many reference distributions. We present two different solution concepts for robust joint chance-constrained optimization problems with these ambiguity sets and non-convex constraint functions. Both concepts rely on solving an approximation problem that is based on well-known smoothing and penalization techniques. On the one side, we consider a classical bundle method together with an approach for finding good starting points. On the other side, we integrate the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent that is able to exploit regularity in the data. On the example of gas networks we compare the two algorithmic concepts for different topologies and two types of mixture ambiguity sets with Gaussian reference distributions and polyhedral and ϕ-divergence based feasible sets for the mixing coefficients. The results show that both solution approaches are well-suited to solve this difficult problem class. Based on the numerical results we provide some general advices for choosing the more efficient algorithm depending on the main challenges of the considered optimization problem. We give an outlook for the applicability of the method in a wider context. AB - Constructing ambiguity sets in distributionally robust optimization is difficult and currently receives increased attention. In this paper, we focus on mixture models with finitely many reference distributions. We present two different solution concepts for robust joint chance-constrained optimization problems with these ambiguity sets and non-convex constraint functions. Both concepts rely on solving an approximation problem that is based on well-known smoothing and penalization techniques. On the one side, we consider a classical bundle method together with an approach for finding good starting points. On the other side, we integrate the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent that is able to exploit regularity in the data. On the example of gas networks we compare the two algorithmic concepts for different topologies and two types of mixture ambiguity sets with Gaussian reference distributions and polyhedral and ϕ-divergence based feasible sets for the mixing coefficients. The results show that both solution approaches are well-suited to solve this difficult problem class. Based on the numerical results we provide some general advices for choosing the more efficient algorithm depending on the main challenges of the considered optimization problem. We give an outlook for the applicability of the method in a wider context. Y1 - 2025 ER - TY - CPAPER U1 - Konferenzveröffentlichung A1 - Disser, Yann A1 - Griesbach, Svenja M. A1 - Klimm, Max A1 - Lutz, Annette T1 - Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem N2 - We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B∈R≥0 is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4ℓ−1)χ,(2^(ℓ+2))/(2^ℓ−1))-approximation for every fixed ℓ∈N. AB - We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B∈R≥0 is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4ℓ−1)χ,(2^(ℓ+2))/(2^ℓ−1))-approximation for every fixed ℓ∈N. KW - incremental maximization KW - competitive analysis KW - prize-collecting Steiner-tree Y1 - 2025 SP - 21 ER - TY - INPR U1 - Preprint A1 - Berrens, Arne A1 - Giesselmann, Jan T1 - A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport N2 - We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error. AB - We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error. KW - cross-diffusion KW - ion transport KW - finite-volume approximation KW - a posteriori error estimates KW - diffusion equation Y1 - 2025 ER - TY - INPR U1 - Preprint A1 - Birke, Gunnar A1 - Engwer, Christian A1 - Giesselmann, Jan A1 - May, Sandra T1 - Error analysis of a first-order DoD cut cell method for 2D unsteady advection N2 - In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the L∞-in-time L2-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results. AB - In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the L∞-in-time L2-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results. KW - cut cell KW - discontinuous Galerkin method KW - DoD Stabilization KW - a priori error estimate KW - unsteady advection Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Kwon, Kiwoong A1 - Lee, Min-Gi T1 - Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system N2 - This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of N particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension d≥2, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing L2 state differences before the formation of δ-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as δ-shocks, vacuums, and shock discontinuities in the weak solutions. AB - This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of N particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension d≥2, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing L2 state differences before the formation of δ-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as δ-shocks, vacuums, and shock discontinuities in the weak solutions. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Topalovic, Antonia A1 - Hante, Falk M. T1 - Stabilizing Model Predictive Control for Generalized Nash Equilibrium Problems using Approximation by α -quasi-GNEPs and Lyapunov End Cost N2 - We study model predictive control (MPC) schemes for non-cooperative dynamic games regarding stabilization. The dynamic games are modelled as generalized Nash equilibrium problems (GNEPs), in which a shared constraint is given as a jointly controlled time-discrete (linear) dynamics. Furthermore, the players’ objectives are interdependent. We present recent results concerning their stabilizing properties using α-quasi- GENP-approximation and terminal conditions in the form of equilibrium endpoint constraints. Moreover, we extend the result towards Lyapunov terminal costs, which is a more general type of terminal condition. Furthermore, we show that a suitable Lyapunov terminal cost can be obtained from a non-game- based MPC scheme. This non-game-based MPC scheme relies on a classical optimal control problem for the aggregated cost. Hence, known results for determining the Lyapunov cost can be applied and carried over to the game-based setting. The theoretical results are complemented by numerical experiments. AB - We study model predictive control (MPC) schemes for non-cooperative dynamic games regarding stabilization. The dynamic games are modelled as generalized Nash equilibrium problems (GNEPs), in which a shared constraint is given as a jointly controlled time-discrete (linear) dynamics. Furthermore, the players’ objectives are interdependent. We present recent results concerning their stabilizing properties using α-quasi- GENP-approximation and terminal conditions in the form of equilibrium endpoint constraints. Moreover, we extend the result towards Lyapunov terminal costs, which is a more general type of terminal condition. Furthermore, we show that a suitable Lyapunov terminal cost can be obtained from a non-game- based MPC scheme. This non-game-based MPC scheme relies on a classical optimal control problem for the aggregated cost. Hence, known results for determining the Lyapunov cost can be applied and carried over to the game-based setting. The theoretical results are complemented by numerical experiments. Y1 - 2025 SP - 6 ER - TY - INPR U1 - Preprint A1 - Aigner, Kevin-Martin A1 - Goerigk, Marc A1 - Hartisch, Michael A1 - Liers, Frauke A1 - Miehlich, Arthur A1 - Rösel, Florian T1 - Feature Selection for Data-Driven Explainable Optimization N2 - Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: a solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to explain a solution. In mathematical optimization, feature selection has received little attention yet. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size. AB - Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: a solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to explain a solution. In mathematical optimization, feature selection has received little attention yet. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size. KW - Feature selection KW - Explainable optimization KW - Data-driven optimization Y1 - 2025 SP - 21 ER - TY - INPR U1 - Preprint A1 - Hildebrand, Robert A1 - Göß, Adrian T1 - Complexity of Integer Programming in Reverse Convex Sets via Boundary Hyperplane Cover N2 - We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, Boundary Hyperplane Cover, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed. AB - We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, Boundary Hyperplane Cover, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Göß, Adrian A1 - Burlacu, Robert A1 - Martin, Alexander T1 - Parabolic Approximation & Relaxation for MINLP N2 - We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems. AB - We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems. KW - mixed-integer nonlinear programming KW - mixed-integer linear programming KW - quadratic approximation KW - global optimization Y1 - 2025 ER - TY - RPRT U1 - Arbeitspapier A1 - Amer, Zeina A1 - Avdzhieva, Ana A1 - Bongarti, Marcelo A1 - Dvurechensky, Pavel A1 - Farrell, Patricio A1 - Gotzes, Uwe A1 - Hante, Falk M. A1 - Karsai, Attila A1 - Kater, Stefan A1 - Liero, Matthias A1 - Spreckelsen, Klaus A1 - Taraz, Johannes A1 - Peschka, Dirk A1 - Plato, Luisa T1 - Modeling Hydrogen Embrittlement for Pricing Degradation in Gas Pipelines N2 - This paper addresses the critical challenge of hydrogen embrittlement in the context of Germany’s transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: (1) We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. (2) Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany’s future energy infrastructure. AB - This paper addresses the critical challenge of hydrogen embrittlement in the context of Germany’s transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: (1) We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. (2) Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany’s future energy infrastructure. Y1 - 2025 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Giesselmann, Jan A1 - Karsai, Attila A1 - Tscherpel, Tabea T1 - Energy-consistent Petrov-Galerkin time discretization of port-Hamiltonian systems N2 - For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov-Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated. AB - For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov-Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated. Y1 - 2024 U6 - https://doi.org/10.5802/smai-jcm.127 DO - https://doi.org/10.5802/smai-jcm.127 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Structure-preserving $H_\infty$ control for port-Hamiltonian systems N2 - We study $H_\infty$ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model. AB - We study $H_\infty$ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model. Y1 - 2023 U6 - https://doi.org/10.1016/j.sysconle.2023.105493 DO - https://doi.org/10.1016/j.sysconle.2023.105493 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Karsai, Attila T1 - Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply N2 - Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property. AB - Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property. Y1 - 2024 U6 - https://doi.org/10.1007/s00498-024-00384-7 DO - https://doi.org/10.1007/s00498-024-00384-7 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Karsai, Attila A1 - Breiten, Tobias A1 - Ramme, Justus A1 - Schulze, Philipp T1 - Passivity Encoding Representations of Nonlinear Systems N2 - Passive systems are characterized by their inability to generate energy internally, providing a powerful tool for modeling physical phenomena. In addition, algebraically encoding passivity in the system description can be advantageous. For this, port-Hamiltonian systems are a prominent approach. Another possibility is writing the system in suitable coordinates. In this article, we investigate the equivalence between passivity and the feasibility of passivity encoding representations, thereby elaborating upon existing results for port-Hamiltonian systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known. AB - Passive systems are characterized by their inability to generate energy internally, providing a powerful tool for modeling physical phenomena. In addition, algebraically encoding passivity in the system description can be advantageous. For this, port-Hamiltonian systems are a prominent approach. Another possibility is writing the system in suitable coordinates. In this article, we investigate the equivalence between passivity and the feasibility of passivity encoding representations, thereby elaborating upon existing results for port-Hamiltonian systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known. Y1 - 2024 U6 - https://doi.org/10.1109/TAC.2025.3576535 DO - https://doi.org/10.1109/TAC.2025.3576535 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Passive feedback control for nonlinear systems N2 - Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this paper, we show that a particular choice of the observer gain gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. In addition, we propose a novel passivity preserving discrete gradient scheme for the time discretization of passive systems. To illustrate our results, we numerically realize the controller using the policy iteration and compare it to a controller where the observer gain is given by the extended Kalman filter. AB - Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this paper, we show that a particular choice of the observer gain gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. In addition, we propose a novel passivity preserving discrete gradient scheme for the time discretization of passive systems. To illustrate our results, we numerically realize the controller using the policy iteration and compare it to a controller where the observer gain is given by the extended Kalman filter. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2502.04987 DO - https://doi.org/10.48550/arXiv.2502.04987 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin A1 - Schuster, Michael T1 - Optimal Neumann Control of the Wave Equation with L1-Control Cost: The Finite-Time Turnpike Property JF - Optimization N2 - The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an 𝐿1-norm control cost with a weight 𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared 𝐿2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the 𝐿1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong 𝐿2-sense. AB - The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an 𝐿1-norm control cost with a weight 𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared 𝐿2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the 𝐿1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong 𝐿2-sense. Y1 - 2024 U6 - https://doi.org/https://doi.org/10.1080/02331934.2024.2373904 DO - https://doi.org/https://doi.org/10.1080/02331934.2024.2373904 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Aigner, Kevin-Martin A1 - Denzler, Sebastian A1 - Liers, Frauke A1 - Pokutta, Sebastian A1 - Sharma, Kartikey T1 - Scenario Reduction for Distributionally Robust Optimization N2 - Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the oroposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors. AB - Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the oroposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors. KW - distributionally robust optimization KW - scenario reduction KW - scenario clustering KW - approximation bounds KW - mixed-integer programming Y1 - 2025 SP - 36 ER - TY - INPR U1 - Preprint A1 - Strubberg, Lea A1 - Lutz, Annette A1 - Börner, Pascal A1 - Pfetsch, Marc A1 - Skutella, Martin A1 - Klimm, Max T1 - Valid Cuts for the Design of Potential-based Flow Networks N2 - The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program. Its non-convexity, due to the poten- tial flow, together with the binary variables, indicating the decision to build a connection, make these problems challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the bi- nary variables. Further, we show that this class of inequalities can be sep- arated in polynomial for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-bound al- gorithm. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks. AB - The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program. Its non-convexity, due to the poten- tial flow, together with the binary variables, indicating the decision to build a connection, make these problems challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the bi- nary variables. Further, we show that this class of inequalities can be sep- arated in polynomial for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-bound al- gorithm. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks. KW - potential based flows KW - topology optimization KW - MINLP Y1 - 2025 SP - 18 ER - TY - INPR U1 - Preprint A1 - Bock de Barillas, Paulina A1 - Hante, Falk A1 - Hintermüller, Michael T1 - Exact relaxation in optimal switching control for the heat equation N2 - We consider an optimal control problem for the heat equation as a prototypical parabolic partial differential equation with a non-convex control mechanism of the form continuous-or-off. We model this fundamental switching mechanism as the product of a classically continuous and a binary control both in the control term of the dynamics and in the objective. A total variation regularization is added to the cost in order to restrict the number of switching times. This renders the problem as a mixed-integer non-linear PDE-constrained problem. We discuss well-posedness of the problem and present an exact relaxation result for a linearized and a trust-region type penalized problem. The exactness result is constructive and provides a way to numerically compute mixed-integer optimal solutions from the optimality conditions of an associated PDE-constrained problem without integer restrictions. It lays a foundation for a new class of sequential relaxation algorithms to solve the considered class of mixed-integer control problems. This is demonstrated numerically by showcasing a descent step in the presence of binary restrictions. AB - We consider an optimal control problem for the heat equation as a prototypical parabolic partial differential equation with a non-convex control mechanism of the form continuous-or-off. We model this fundamental switching mechanism as the product of a classically continuous and a binary control both in the control term of the dynamics and in the objective. A total variation regularization is added to the cost in order to restrict the number of switching times. This renders the problem as a mixed-integer non-linear PDE-constrained problem. We discuss well-posedness of the problem and present an exact relaxation result for a linearized and a trust-region type penalized problem. The exactness result is constructive and provides a way to numerically compute mixed-integer optimal solutions from the optimality conditions of an associated PDE-constrained problem without integer restrictions. It lays a foundation for a new class of sequential relaxation algorithms to solve the considered class of mixed-integer control problems. This is demonstrated numerically by showcasing a descent step in the presence of binary restrictions. Y1 - 2025 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin T1 - New Lyapunov functions for systems with source terms JF - Control and Cybernetics N2 - Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of 2 × 2 systems of balance laws. We present cases connected with the study of the limit of stabilizability, where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights. Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set, for which the system can be stabilized becomes substantially smaller. We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations. AB - Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of 2 × 2 systems of balance laws. We present cases connected with the study of the limit of stabilizability, where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights. Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set, for which the system can be stabilized becomes substantially smaller. We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations. KW - Lyapunov Function KW - Exponential Weight KW - Hyperbolic Weight KW - Feedback law KW - Stabilization Y1 - 2025 U6 - https://doi.org/https://doi.org/10.2478/candc-2024-0008 DO - https://doi.org/https://doi.org/10.2478/candc-2024-0008 VL - 53 IS - 1 SP - 163 EP - 187 S1 - 25 ER - TY - INPR U1 - Preprint A1 - Breiten, Tobias A1 - Burela, Shubhaditya A1 - Schulze, Philipp T1 - Optimal Control for a Class of Linear Transport-Dominated Systems via the Shifted Proper Orthogonal Decomposition N2 - Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. AB - Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. Y1 - 2025 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Schuster, Michael A1 - Gugat, Martin A1 - Sokolowski, Jan T1 - The Location Problem for Compressor Stations in Pipeline Networks JF - Mathematics and Mechanics of Complex Systems N2 - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related to the gas dynamics that governs the network flow. That results in nonconvex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, we consider the problem of finding the optimal location for the control on the network such that the control cost is minimal and the gas pressure stays within given bounds. In the deterministic setting, we present explicit bounds for the pipe length and the inlet pressure such that a unique optimal compressor location with minimal control cost exists. In the probabilistic setting, we give an existence result for the optimal compressor location and discuss the uniqueness of the solution depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. We further present the problem of finding optimal compressor locations on networks including the number of compressor stations as a variable. Results for the existence of optimal locations on a graph in both the deterministic and the probabilistic setting are presented, and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example on a diamond graph demonstrating that the minimal number of compressor stations is not necessarily equal to the optimal number of compressor stations. AB - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related to the gas dynamics that governs the network flow. That results in nonconvex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, we consider the problem of finding the optimal location for the control on the network such that the control cost is minimal and the gas pressure stays within given bounds. In the deterministic setting, we present explicit bounds for the pipe length and the inlet pressure such that a unique optimal compressor location with minimal control cost exists. In the probabilistic setting, we give an existence result for the optimal compressor location and discuss the uniqueness of the solution depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. We further present the problem of finding optimal compressor locations on networks including the number of compressor stations as a variable. Results for the existence of optimal locations on a graph in both the deterministic and the probabilistic setting are presented, and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example on a diamond graph demonstrating that the minimal number of compressor stations is not necessarily equal to the optimal number of compressor stations. KW - gas network KW - compressor control KW - Weber problem KW - uncertain boundary data KW - non convex mixed integer stochastic problem Y1 - 2024 U6 - https://doi.org/10.2140 DO - https://doi.org/10.2140 VL - 12 IS - 4 SP - 507 EP - 546 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Filipkovska, Maria T1 - Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks N2 - Conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations have been obtained. An example demonstrating the application of the obtained results has been considered. Isothermal models of gas networks have been proposed as applications. AB - Conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations have been obtained. An example demonstrating the application of the obtained results has been considered. Isothermal models of gas networks have been proposed as applications. KW - nonregular differential-algebraic equation KW - degenerate differential equation KW - singular pencil KW - gas network KW - global solvability Y1 - 2023 U6 - https://doi.org/https://doi.org/10.15407/mag19.04.719 DO - https://doi.org/https://doi.org/10.15407/mag19.04.719 VL - 19 IS - 4 SP - 719 EP - 765 ER - TY - CPAPER U1 - Konferenzveröffentlichung A1 - Börner, Pascal A1 - Pfetsch, Marc E. A1 - Ulbrich, Stefan T1 - Modeling and optimization of gas mixtures on networks N2 - This paper presents a model for the mixture of gases on networks in the stationary case. The model is based on an equation of state for the mixture, the stationary isothermal Euler equations and coupling conditions for the flow and mixture. The equation of state or pressure law is based on the change of the speed of sound in a mixture of gases. We use this model to solve stationary gas flow problems to global optimality on large networks and present computational results. AB - This paper presents a model for the mixture of gases on networks in the stationary case. The model is based on an equation of state for the mixture, the stationary isothermal Euler equations and coupling conditions for the flow and mixture. The equation of state or pressure law is based on the change of the speed of sound in a mixture of gases. We use this model to solve stationary gas flow problems to global optimality on large networks and present computational results. Y1 - 2024 SP - 6 ER - TY - INPR U1 - Preprint A1 - Bernhard, Daniela A1 - Liers, Frauke A1 - Stingl, Michael T1 - Robust chance-constrained optimization with discrete distributions N2 - Typically, probability distributions that generate uncertain parameters cannot be measured exactly in practice. As a remedy, distributional robustness determines optimized decisions that are protected in a robust fashion against all probability distributions in some appropriately chosen ambiguity set. In this work, we consider robust joint chance-constrained optimization problems and focus on discrete probability distributions. Many methods for this kind of problems study convex or even linear constraint functions. In contrast, we introduce a practically efficient scenario-based bundle method without convexity assumptions on the constraint functions. We start by deriving an approximation problem to the original robust chance-constrained version by using smoothing and penalization techniques that build on our former work on chance-constrained optimization. Our convergence results with respect to the smoothing approximation and well-known results for penalty approximations suggest replacing the original problem with the approximation problem for large smoothing and penalty parameters. Our scenario-based bundle method starts by solving the approximation problem with a bundle method, and then uses the bundle solution to decide which scenarios to include in a scenario-expanded formulation. This formulation is a standard nonlinear optimization problem. Our approach is guaranteed to find feasible solutions. Furthermore, in the numerical experiments on real-world gas transport problems with uncertain demands, we mostly find globally optimal solutions. Comparing these results to the classical robust reformulations for ambiguity sets consisting of confidence intervals and Wasserstein balls, we observe that the scenario-based bundle method typically outperforms solving the classical reformulation directly. AB - Typically, probability distributions that generate uncertain parameters cannot be measured exactly in practice. As a remedy, distributional robustness determines optimized decisions that are protected in a robust fashion against all probability distributions in some appropriately chosen ambiguity set. In this work, we consider robust joint chance-constrained optimization problems and focus on discrete probability distributions. Many methods for this kind of problems study convex or even linear constraint functions. In contrast, we introduce a practically efficient scenario-based bundle method without convexity assumptions on the constraint functions. We start by deriving an approximation problem to the original robust chance-constrained version by using smoothing and penalization techniques that build on our former work on chance-constrained optimization. Our convergence results with respect to the smoothing approximation and well-known results for penalty approximations suggest replacing the original problem with the approximation problem for large smoothing and penalty parameters. Our scenario-based bundle method starts by solving the approximation problem with a bundle method, and then uses the bundle solution to decide which scenarios to include in a scenario-expanded formulation. This formulation is a standard nonlinear optimization problem. Our approach is guaranteed to find feasible solutions. Furthermore, in the numerical experiments on real-world gas transport problems with uncertain demands, we mostly find globally optimal solutions. Comparing these results to the classical robust reformulations for ambiguity sets consisting of confidence intervals and Wasserstein balls, we observe that the scenario-based bundle method typically outperforms solving the classical reformulation directly. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation N2 - We study a state estimation problem for a 2x2 linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term. AB - We study a state estimation problem for a 2x2 linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Wiertz, Ann-Kathrin A1 - Walther, Andrea A1 - Zöttl, Gregor T1 - Strategic Retailers in the Energy Sector N2 - We propose a general framework which allows to analyze the strategic interaction of retail companies, where customers choose retail contracts over a longer period of time based on price and non-price characteristics of retail contracts. We allow for many, possibly asymmetric retailers which can offer fixed price tariffs or dynamic prices, as typically observed in energy markets. Our framework considers uncertainties and allows for price-responsive consumption choices of customers. We analytically characterize all resulting market equilibria for the general asymmetric setting. Based on those results we then propose a solution algorithm which is capable to determine all resulting equilibria. For the case of symmetric retailers we provide analytical comparisons of the different tariff structures. To show the applicability of our framework and our algorithm to real-world instances, we calibrate it to data of the German retail electricity market. Our results show, that firms profits remain unchanged but consumer surplus and welfare increase when switching from fixed price tariffs to real-time pricing. This effect is more pronounced under higher wholesale price fluctuations. Finally we also propose a surrogate, reduced order model, which is shown to be equivalently capable to quantify the welfare difference of the different tariffs. AB - We propose a general framework which allows to analyze the strategic interaction of retail companies, where customers choose retail contracts over a longer period of time based on price and non-price characteristics of retail contracts. We allow for many, possibly asymmetric retailers which can offer fixed price tariffs or dynamic prices, as typically observed in energy markets. Our framework considers uncertainties and allows for price-responsive consumption choices of customers. We analytically characterize all resulting market equilibria for the general asymmetric setting. Based on those results we then propose a solution algorithm which is capable to determine all resulting equilibria. For the case of symmetric retailers we provide analytical comparisons of the different tariff structures. To show the applicability of our framework and our algorithm to real-world instances, we calibrate it to data of the German retail electricity market. Our results show, that firms profits remain unchanged but consumer surplus and welfare increase when switching from fixed price tariffs to real-time pricing. This effect is more pronounced under higher wholesale price fluctuations. Finally we also propose a surrogate, reduced order model, which is shown to be equivalently capable to quantify the welfare difference of the different tariffs. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Giannakopoulos, Yiannis A1 - Hahn, Johannes T1 - Discrete Single-Parameter Optimal Auction Design N2 - We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over "finite" supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a linear combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction. AB - We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over "finite" supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a linear combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction. Y1 - 2024 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation N2 - The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided. AB - The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided. KW - Turnpike Property KW - Wave Equation KW - Optimal Control KW - Spectral Method KW - Network Optimum Design Y1 - 2024 VL - J. Geom. Anal. IS - 34 AR - 273 S2 - 273 SP - 60 ER - TY - INPR U1 - Preprint A1 - Lange, Christian T1 - Modeling and Optimal Control of the Flow of a Gas Mixture N2 - We consider the Euler equations for a pipeline flow of a mixture of two gases. An important application is hydrogen blending. Existence and uniqueness of semi-global solutions is shown and possible boundary conditions are analyzed. Secondly, we consider classes of associated optimal control problems and show existence of solutions. AB - We consider the Euler equations for a pipeline flow of a mixture of two gases. An important application is hydrogen blending. Existence and uniqueness of semi-global solutions is shown and possible boundary conditions are analyzed. Secondly, we consider classes of associated optimal control problems and show existence of solutions. KW - Euler Equations KW - quasilinear hyperbolic system KW - gas transport modeling KW - hydrogen blending Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Gugat, Martin A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - Location Problem for Compressor Stations in Pipeline Networks N2 - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related with the gas dynamics that governs the network flow. That results in non-convex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, the problem of finding the optimal location for the control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered. In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting, an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. Results for the existence of optimal locations on a graph in both, the deterministic and the probabilistic setting, are presented and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings. AB - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related with the gas dynamics that governs the network flow. That results in non-convex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, the problem of finding the optimal location for the control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered. In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting, an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. Results for the existence of optimal locations on a graph in both, the deterministic and the probabilistic setting, are presented and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings. KW - Gas Networks KW - Compressor Control KW - Weber Problem KW - Optimal Location KW - Uncertain Boundary Data Y1 - 2024 SP - 34 ER - TY - INPR U1 - Preprint A1 - Schuster, Michael T1 - On the Convergence of Optimization Problems with Kernel Density Estimated Probabilistic Constraints N2 - Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields, even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluate corresponding probabilities. In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity. The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems. AB - Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields, even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluate corresponding probabilities. In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity. The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems. KW - Probabilistic Constrained Optimization KW - Stochastic Optimization KW - Chance Constraints KW - Kernel Density Estimation KW - Convergence Analysis Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Hante, Falk M. A1 - Schmidt, Martin A1 - Topalovic, Antonia T1 - Stabilizing GNEP-Based Model Predictive Control: α-Quasi-GNEPs and End Constraints N2 - We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP) in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop locally around the steady state if stabilizing terminal constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing terminal constraints. The theoretical results are complemented by numerical experiments. AB - We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP) in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop locally around the steady state if stabilizing terminal constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing terminal constraints. The theoretical results are complemented by numerical experiments. KW - Model predictive control KW - Non-cooperative distributed control KW - Closed-loop stability KW - Generalized Nash equilibrium problems Y1 - 2024 SP - 30 ER - TY - INPR U1 - Preprint A1 - Kuchlbauer, Martina T1 - Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems N2 - We consider mixed-integer nonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex. We propose a robust optimization method that requires no certain structure of the adversarial problem, but only approximate worst-case evaluations. The method integrates a bundle method, for continuous subproblems, into an outer approximation approach. We prove that our algorithm converges and finds an approximately robust optimal solution and propose robust gas transport as a suitable application. AB - We consider mixed-integer nonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex. We propose a robust optimization method that requires no certain structure of the adversarial problem, but only approximate worst-case evaluations. The method integrates a bundle method, for continuous subproblems, into an outer approximation approach. We prove that our algorithm converges and finds an approximately robust optimal solution and propose robust gas transport as a suitable application. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Thürauf, Johannes A1 - Grübel, Julia A1 - Schmidt, Martin T1 - Adjustable Robust Nonlinear Network Design under Demand Uncertainties N2 - We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations. AB - We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations. KW - Robust Optimization KW - Nonlinear Flows KW - Potential-based Networks KW - Demand Uncertainties KW - Mixed-integer Nonlinear Optimization Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Wilka, Hendrik A1 - Lang, Jens T1 - Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks N2 - High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods. AB - High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods. Y1 - 2024 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution N2 - Method-of-lines discretizations are demanding test problems for stiff inte- gration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To over- come these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numeri- cally observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems. AB - Method-of-lines discretizations are demanding test problems for stiff inte- gration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To over- come these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numeri- cally observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems. Y1 - 2024 U6 - https://doi.org/https://doi.org/10.1007/s10957-022-02154-4 DO - https://doi.org/https://doi.org/10.1007/s10957-022-02154-4 VL - Journal of Optimization Theory and Applications IS - Vol. 196 SP - 1106 EP - 1118 S1 - 13 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Strelow, Erik Laurin A1 - Gerisch, Alf A1 - Lang, Jens A1 - Pfetsch, Marc E. T1 - Physics-Informed Neural Networks: A Case Study for Gas Transport Problems N2 - Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method. AB - Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method. Y1 - 2024 VL - Journal of Computational Physics IS - Vol. 481 SP - 112041 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems N2 - This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84)-stable method with nodes symmetric in [0,1] to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability. AB - This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84)-stable method with nodes symmetric in [0,1] to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability. Y1 - 2024 U6 - https://doi.org/https://doi.org/10.3390/a15090310 DO - https://doi.org/https://doi.org/10.3390/a15090310 VL - Algorithms IS - Vol. 15 AR - 310 S2 - 310 ER - TY - INPR U1 - Preprint A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Breiten, Tobias A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Hiller, Benjamin A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview (Extended English Version) N2 - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. AB - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. Y1 - 2024 SP - 54 ER - TY - INPR U1 - Preprint A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control N2 - It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of practical interest. We prove convergence for s-stage methods, for instance, order s for the state variables even if the adjoint method and the control satisfy the conditions for order s-1, only. Numerical tests show the expected order of convergence for the new Peer triplets. AB - It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of practical interest. We prove convergence for s-stage methods, for instance, order s for the state variables even if the adjoint method and the control satisfy the conditions for order s-1, only. Numerical tests show the expected order of convergence for the new Peer triplets. Y1 - 2024 VL - http://arxiv.org/abs/2303.18180 ER - TY - INPR U1 - Preprint A1 - Graser, Gertrud A1 - Kreimeier, Timo A1 - Walther, Andrea T1 - Solving Linear Generalized Nash Games Using an Active Signature Method N2 - We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems. AB - We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Alldredge, Graham A1 - Frank, Martin A1 - Giesselmann, Jan T1 - On the convergence of the regularized entropy-based moment method for kinetic equations N2 - The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal. AB - The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal. Y1 - 2023 U6 - https://doi.org/https://doi.org/10.5802/smai-jcm.93 DO - https://doi.org/https://doi.org/10.5802/smai-jcm.93 VL - 9 SP - 1-29 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Kolbe, Niklas T1 - A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model N2 - We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ. AB - We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ. KW - Keller-Segel KW - chemotaxis; KW - nonlinear diffusion KW - finite volume scheme KW - a posteriori error analysis Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Kwon, Kiwoong T1 - A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model N2 - We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results. AB - We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results. KW - Keller-Segel KW - chemotaxis KW - nonlinear diffusion KW - discontinuous Galerkin scheme KW - a posteriori error analysis Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Giesselmann, Jan A1 - Krupa, Sam T1 - Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems N2 - In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments. AB - In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments. KW - Conservation laws KW - entropy conditions KW - entropy solutions KW - shocks, KW - a posteriori error estimates Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Egger, Herbert A1 - Giesselmann, Jan T1 - Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization N2 - We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem. AB - We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem. KW - gas transport KW - doubly nonlinear parabolic problems KW - relative entropy estimates KW - exponential stability KW - structure preserving discretization Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Gugat, Martin A1 - Giesselmann, Jan T1 - An Observer for pipeline flow with hydrogen blending in gas networks: exponential synchronization N2 - We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the L2-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights. AB - We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the L2-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Bernhard, Daniela A1 - Liers, Frauke A1 - Stingl, Michael A1 - Uihlein, Andrian T1 - A Gradient-Based Method for Joint Chance-Constrained Optimization with Continuous Distributions N2 - The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances. AB - The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances. Y1 - 2024 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Geiersbach, Caroline A1 - Henrion, René T1 - Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form N2 - In this paper, we discuss optimality conditions for optimization problems {involving} random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution. AB - In this paper, we discuss optimality conditions for optimization problems {involving} random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution. Y1 - 2024 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Geiersbach, Caroline A1 - Henrion, René A1 - Pérez-Aros, Pedro T1 - Numerical solution of an optimal control problem with probabilistic and almost sure state constraints N2 - We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated. AB - We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated. Y1 - 2024 ER - TY - INPR U1 - Preprint A1 - Hante, Falk A1 - Kuchler, Christian T1 - Indirect methods for optimal control of parabolic hybrid PDE-dynamical / switching systems using relaxation N2 - We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear parabolic partial differential equations (PDEs) including a state-dependent control-regime switching mechanism. We state an equivalent mixed-integer formulation featuring vanishing constraints (VCs) arising from methods of disjunctive programming. We embed the problem into the class of equilibrium constraints (ECs) by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding (SUR) strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach for the VCs. After a theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on the proposed approach. AB - We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear parabolic partial differential equations (PDEs) including a state-dependent control-regime switching mechanism. We state an equivalent mixed-integer formulation featuring vanishing constraints (VCs) arising from methods of disjunctive programming. We embed the problem into the class of equilibrium constraints (ECs) by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding (SUR) strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach for the VCs. After a theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on the proposed approach. Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Grimm, Veronika A1 - Grübel, Julia A1 - Schmidt, Martin A1 - Schwartz, Alexandra A1 - Wiertz, Ann-Kathrin A1 - Zöttl, Gregor T1 - On a Tractable Single-Level Reformulation of a Multilevel Model of the European Entry-Exit Gas Market with Market Power N2 - We propose a framework that allows to quantitatively analyze the interplay of the different agents involved in gas trade and transport in the context of the European entry-exit system. Previous contributions have focused on the case of perfectly competitive buyers and sellers of gas, which allows to replace the respective market equilibrium problem by a single welfare maximization problem. Our novel framework considers the mathematically more challenging case of a monopolistic and thus strategic gas seller. In this framework, the objective functions of the gas sellers and buyers cannot be aggregated into a common objective function, which is why a multilevel formulation is necessary to accurately capture the sequential nature of the decisions taken. For this setup, we derive sufficient conditions that allow for reformulating the challenging four-level model as a computationally tractable single-level reformulation. We prove the correctness of this reformulation and use it for solving several test instances to illustrate the applicability of our approach. AB - We propose a framework that allows to quantitatively analyze the interplay of the different agents involved in gas trade and transport in the context of the European entry-exit system. Previous contributions have focused on the case of perfectly competitive buyers and sellers of gas, which allows to replace the respective market equilibrium problem by a single welfare maximization problem. Our novel framework considers the mathematically more challenging case of a monopolistic and thus strategic gas seller. In this framework, the objective functions of the gas sellers and buyers cannot be aggregated into a common objective function, which is why a multilevel formulation is necessary to accurately capture the sequential nature of the decisions taken. For this setup, we derive sufficient conditions that allow for reformulating the challenging four-level model as a computationally tractable single-level reformulation. We prove the correctness of this reformulation and use it for solving several test instances to illustrate the applicability of our approach. KW - Multilevel optimization KW - Reformulations KW - Gas markets KW - Market power Y1 - 2023 SP - 31 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Shyshkanova, Ganna A1 - Zaytseva, Tetyana A1 - Zhushman, V A1 - Levchenko, Ntaliia A1 - Korotunova, Olena T1 - Solving three-dimensional contact problems for foundation design in green building N2 - Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy. AB - Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy. Y1 - 2023 U6 - https://doi.org/10.1088/1742-6596/2609/1/012001 DO - https://doi.org/10.1088/1742-6596/2609/1/012001 ER - TY - THES U1 - Dissertation oder Habilitation A1 - Krug, Richard T1 - Decomposition Methods for Time-Dependent Mixed-Integer Nonlinear Optimization Problems on Graphs N2 - Decomposition can be the method of choice to deal with optimization problems that contain hard to solve model structures or that are of large scale. The main idea is to decompose the problematic aspects of the problem into multiple smaller blocks that can be solved more easily. Here, the challenge is to combine the single pieces to a solution that is not only feasible but maybe even optimal for the original problem. In many cases, this can be done by introducing an iteration that eventually converges to a desired solution. In this cumulative dissertation, we present several iterative decomposition methods that are tailored to different types of optimization models and use distinct approaches to split up the problems. Our main motivation for this originates from the optimization of gas transport networks, where we encounter partial differential equations as well as discrete control decisions. Additionally, we engage in the related field of district heating network optimization to study the challenges arising from large-scale and fully discretized systems as well as undesirable model features such as, e.g., complementarity constraints. Here, we introduce two temperature mixing models that are well suited for optimization and a number of techniques to speed up the solution process, which are applied in numerical experiments. As a next step, we develop an iterative time-domain decomposition method that is applied to optimal control problems subject to semilinear hyperbolic systems of partial differential equations. For this, we derive first-order optimality conditions that are then split using a non-overlapping decomposition of the time horizon. We exploit the fact that the resulting systems have a primal interpretation as so-called virtual control problems. We prove the convergence of the iterative method and develop a posteriori error estimates. Later, we extend the scheme to systems of ordinary differential equations with mixed- integer controls by using Pontryagin’s maximum principle. We again show the convergence and conduct a numerical case study. Moreover, we use a consensus-based version of the classic penalty alternating direction method to solve tailored reformulations of transient gas network problems that allow us to minimize the number of coupling constraints between sub-problems. Here, we utilize the quasi-separable structure of the network to decompose it into sub-networks with more desirable properties. We also discuss different decomposition strategies and test them in a numerical case study. Finally, we present a successive linear relaxation method for mixed-integer nonlinear problems with multivariate Lipschitz continuous nonlinearities. The distinguishing feature of this algorithm is that it exploits no properties of the nonlinearities besides the Lipschitz constants. Therefore, the method is applicable for problems with non-convex or even non-differentiable constraints. The nonlinearities do not even need to be given in a closed form, which allows us to integrate black-box constraints into the model. We prove that the algorithm converges to an approximate global optimum and we provide a worst-case estimate for the number of iterations. The iterative method is applied to stationary gas transport problems, where implicitly given solutions of the differential equations are modeled via black-box constraints. AB - Decomposition can be the method of choice to deal with optimization problems that contain hard to solve model structures or that are of large scale. The main idea is to decompose the problematic aspects of the problem into multiple smaller blocks that can be solved more easily. Here, the challenge is to combine the single pieces to a solution that is not only feasible but maybe even optimal for the original problem. In many cases, this can be done by introducing an iteration that eventually converges to a desired solution. In this cumulative dissertation, we present several iterative decomposition methods that are tailored to different types of optimization models and use distinct approaches to split up the problems. Our main motivation for this originates from the optimization of gas transport networks, where we encounter partial differential equations as well as discrete control decisions. Additionally, we engage in the related field of district heating network optimization to study the challenges arising from large-scale and fully discretized systems as well as undesirable model features such as, e.g., complementarity constraints. Here, we introduce two temperature mixing models that are well suited for optimization and a number of techniques to speed up the solution process, which are applied in numerical experiments. As a next step, we develop an iterative time-domain decomposition method that is applied to optimal control problems subject to semilinear hyperbolic systems of partial differential equations. For this, we derive first-order optimality conditions that are then split using a non-overlapping decomposition of the time horizon. We exploit the fact that the resulting systems have a primal interpretation as so-called virtual control problems. We prove the convergence of the iterative method and develop a posteriori error estimates. Later, we extend the scheme to systems of ordinary differential equations with mixed- integer controls by using Pontryagin’s maximum principle. We again show the convergence and conduct a numerical case study. Moreover, we use a consensus-based version of the classic penalty alternating direction method to solve tailored reformulations of transient gas network problems that allow us to minimize the number of coupling constraints between sub-problems. Here, we utilize the quasi-separable structure of the network to decompose it into sub-networks with more desirable properties. We also discuss different decomposition strategies and test them in a numerical case study. Finally, we present a successive linear relaxation method for mixed-integer nonlinear problems with multivariate Lipschitz continuous nonlinearities. The distinguishing feature of this algorithm is that it exploits no properties of the nonlinearities besides the Lipschitz constants. Therefore, the method is applicable for problems with non-convex or even non-differentiable constraints. The nonlinearities do not even need to be given in a closed form, which allows us to integrate black-box constraints into the model. We prove that the algorithm converges to an approximate global optimum and we provide a worst-case estimate for the number of iterations. The iterative method is applied to stationary gas transport problems, where implicitly given solutions of the differential equations are modeled via black-box constraints. Y1 - 2023 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Ouanes, Nesrine A1 - González Grandón, Tatiana A1 - Heitsch, Holger A1 - Henrion, René T1 - Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints N2 - In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In many countries with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: stochastic solar power and demand forecast errors; absolute uncertain national grid outage onset times; and outages duration subjected to statistical analysis. These uncertainties pose new challenges to the classical power system’s operation tasks. Two alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC) and a purely deterministic dispatch model. The JCC model has the capability to address all four uncertainties, while the ICC covers only three of them, overlooking the uncertainty about the outage duration. In contrast, the purely deterministic model completely ignores any uncertain parameters. We illustrate the three models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. Results show how the dispatch is modified across the models to plan the battery and diesel reserves in the chance-constrained models, with the reserves in the JCC being larger than in the ICC model. In comparison between all models, we prove that the JCC model offers the most robust results, since it can handle uncertainties about forecasting errors, on the one hand, and grid outages, on the other. The results also show that the decrease in profits due to the hedging with reserves kept in the MG is significantly small compared to the high level of reliability reached and the potential load shedding that could be avoided in the case of an outage. AB - In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In many countries with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: stochastic solar power and demand forecast errors; absolute uncertain national grid outage onset times; and outages duration subjected to statistical analysis. These uncertainties pose new challenges to the classical power system’s operation tasks. Two alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC) and a purely deterministic dispatch model. The JCC model has the capability to address all four uncertainties, while the ICC covers only three of them, overlooking the uncertainty about the outage duration. In contrast, the purely deterministic model completely ignores any uncertain parameters. We illustrate the three models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. Results show how the dispatch is modified across the models to plan the battery and diesel reserves in the chance-constrained models, with the reserves in the JCC being larger than in the ICC model. In comparison between all models, we prove that the JCC model offers the most robust results, since it can handle uncertainties about forecasting errors, on the one hand, and grid outages, on the other. The results also show that the decrease in profits due to the hedging with reserves kept in the MG is significantly small compared to the high level of reliability reached and the potential load shedding that could be avoided in the case of an outage. Y1 - 2023 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Shyshkanova, Ganna A1 - Walther, Andrea T1 - Optimization of a punch shape with a doubly connected contact domain N2 - The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined. AB - The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined. Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Goerigk, Marc A1 - Kurtz, Jannis A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Connections between Robust and Bilevel Optimization N2 - Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (static robust problems with and without decision-dependence of their uncertainty sets, worst-case regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields - in particular to use both theory and algorithms from one field in the other and vice versa. AB - Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (static robust problems with and without decision-dependence of their uncertainty sets, worst-case regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields - in particular to use both theory and algorithms from one field in the other and vice versa. KW - Bilevel optimization KW - Robust optimization KW - Reformulations Y1 - 2023 SP - 22 ER - TY - INPR U1 - Preprint A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Topological derivative method for control of wave equation on networks N2 - The dynamical, boundary optimal control problems on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation on the graph. The boundary control problem by Neumann conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented. AB - The dynamical, boundary optimal control problems on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation on the graph. The boundary control problem by Neumann conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented. KW - distributed parameter system KW - optimal control KW - shape optimization KW - topological derivative KW - network modelling Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Kannan, Aswin A1 - Kreimeier, Timo A1 - Walther, Andrea T1 - On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Hante, Falk A1 - Kuchler, Christian T1 - An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs N2 - We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method. After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework. AB - We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method. After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework. Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Göß, Adrian A1 - Martin, Alexander A1 - Pokutta, Sebastian A1 - Sharma, Kartikey T1 - Norm-induced Cuts: Optimization with Lipschitzian Black-box Functions N2 - Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter. We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the “shape” of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example. AB - Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter. We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the “shape” of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example. KW - Global Optimization KW - Lipschitz Optimization KW - Black-box Optimization KW - Derivative-free Optimization Y1 - 2023 ER - TY - INPR U1 - Preprint A1 - Bongarti, Marcelo A1 - Hintermüller, T1 - Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network N2 - The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. AB - The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. KW - optimal boundary control KW - gas dynamics KW - gas networks KW - isothermal Euler equation KW - compressible fluid dynamics Y1 - 2023 ER - TY - CPAPER U1 - Konferenzveröffentlichung A1 - Gugat, Martin A1 - Schuster, Michael T1 - Max-p optimal boundary control of gas flow N2 - In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers’ demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline metal. AB - In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers’ demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline metal. KW - gas pipeline flow KW - nodal control KW - hyperbolic differential equation KW - random demand KW - state constraints Y1 - 2022 U6 - https://doi.org/https://doi.org/10.15495/EPub_UBT_00006809 DO - https://doi.org/https://doi.org/10.15495/EPub_UBT_00006809 VL - Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems Bayreuth, Germany, 12-16 September 2022 SP - 4 ER - TY - JOUR U1 - Wissenschaftlicher Artikel A1 - Gugat, Martin A1 - Lazar, Martin T1 - Turnpike Properties for Partially Uncontrollable Systems N2 - We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework. AB - We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework. KW - Measure Turnpike KW - Averaged Control KW - LQ optimal control problem KW - Infinite-time admissibility KW - Turnpike phenomenon Y1 - 2023 VL - Automatica IS - 149 ER - TY - INPR U1 - Preprint A1 - Schuster, Michael A1 - Sakamoto, Noboru T1 - A Turnpike Result for Optimal Boundary Control Problems with the Transport Equation under Uncertainty N2 - In this paper we analyze the turnpike phenomenon for optimal boundary control problems with a linear transport equation with source term. The convex objective function depends on the boundary traces of the transport equation and is strictly convex with respect to the boundary control. We show an integral turnpike result for an optimal Dirichlet boundary control problem in the sense that if the time horizon goes to infinity, then the dynamic optimal control converges to the corresponding steady state optimal control. The novelty of this work is two-sided. On the one hand, even if turnpike results for this kind of optimal boundary control problem already exist, we present a new direct proof without using adjoint calculus that leads to sharper estimates. On the other hand we consider uncertainty in the initial data and/or in the source term. We show that the integral turnpike result also holds considering uncertainty. Throughout the paper we use numerical examples to illustrate the results. AB - In this paper we analyze the turnpike phenomenon for optimal boundary control problems with a linear transport equation with source term. The convex objective function depends on the boundary traces of the transport equation and is strictly convex with respect to the boundary control. We show an integral turnpike result for an optimal Dirichlet boundary control problem in the sense that if the time horizon goes to infinity, then the dynamic optimal control converges to the corresponding steady state optimal control. The novelty of this work is two-sided. On the one hand, even if turnpike results for this kind of optimal boundary control problem already exist, we present a new direct proof without using adjoint calculus that leads to sharper estimates. On the other hand we consider uncertainty in the initial data and/or in the source term. We show that the integral turnpike result also holds considering uncertainty. Throughout the paper we use numerical examples to illustrate the results. KW - Turnpike KW - Boundary Control KW - Transport Equation KW - Random Boundary Data Y1 - 2023 ER -