TY - JOUR 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. Y1 - 2024 U6 - https://doi.org/10.48550 VL - Geocombinatorics Quarterly IS - Volume XXXIV: 2024 ER - TY - CHAP 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. Y1 - 2024 ER - TY - CHAP 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. Y1 - 2025 ER - TY - JOUR 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. 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 ER - TY - JFULL 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. KW - gas network optimization KW - gas mixing KW - MINLP KW - global optimization Y1 - ER - TY - JOUR A1 - Henrion, René A1 - Schmidt, Martin T1 - Chance-Constrained Linear Complementarity Problems N2 - We study linear complementarity problems (LCPs) under uncer- tainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relax- ations, 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 con- vexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small case study regarding a Cournot oligopoly among energy producers, we finally illustrate the applicability of our theoretical findings. KW - Linear complementarity problems KW - Chance constraints KW - Existence KW - Convexity KW - Optimality conditions Y1 - 2026 ER - TY - INPR A1 - Breiten, Tobias A1 - Karsai, Attila A1 - Mehrmann, Volker A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Tscherpel, Tabea A1 - Hiller, Benjamin A1 - Morandin, Riccardo A1 - Tischendorf, Caren 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. Y1 - 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 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. Y1 - ER - TY - INPR 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. KW - Navier-Stokes KW - blow-up KW - a posteriori estimates KW - critical space KW - reconstruction Y1 - ER - TY - INPR 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. KW - Blended Gas Flow KW - Compressibility Factor KW - z-Factor KW - Real Gas KW - Steady States Y1 - 2026 ER - TY - JOUR 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. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - Lipschitz continuity KW - necessary optimality conditions Y1 - 2025 U6 - https://doi.org/10.2478/udt-2025-0011 VL - 20 IS - 1 SP - 35 EP - 63 ER - TY - JOUR 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. KW - chance constraints KW - continuous stochastic gradient KW - spheric-radial decomposition Y1 - ER - TY - THES 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. 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. Y1 - 2025 ER - TY - JOUR 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. 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 VL - 556 IS - 1 ER - TY - CHAP 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. Y1 - 2026 PB - Springer-Nature ER - TY - JOUR 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. KW - boundary stabilization KW - feedback law KW - varying delay KW - quasi-linear hyperbolic system KW - boundary control Y1 - 2026 U6 - https://doi.org/10.1137/24M1648570 VL - 63 IS - SIAM Journal on Control and Optimization SP - 452 EP - 471 ER - TY - INPR 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. Y1 - 2025 ER - TY - INPR 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. KW - networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control Y1 - 2025 ER - TY - INPR 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. Y1 - ER - TY - INPR 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. KW - hyperbolic systems of conservation laws, shock curves, generalized riemann problem, optimal control, variational calculus Y1 - 2025 ER - TY - INPR 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. KW - robust optimization KW - Bayesian optimization KW - online learning KW - solar cell performance Y1 - 2025 ER -