TY - INPR A1 - Klimm, Max A1 - Pfetsch, Marc A1 - Raber, Rico A1 - Skutella, Martin T1 - Approximation of Binary Second Order Cone Programs of Packing Type N2 - This paper considers binary second order cone programs of packing type where a linear objective is optimized under m second order cone packing constraints and all decision variables are binary. We show that when m is part of the input, these problems cannot be approximated within a factor of 1/(m + 1)^(1−ε) for any ε > 0, unless P = NP. We then propose approximation algorithms based on different algorithmic principles that almost match this approximation factor: a pipage rounding technique that solves fractional relaxations of the problems and modifies the solutions so that few fractional variables remain, a greedy approach, and a randomized rounding technique. While all algorithms have similar theoretical approximation guarantees in the order of 1/m, we also test the algorithms on realistic instances that arise in the context of gas transportation networks. This empirical study reveals in particular that taking the best of the proposed algorithms produces highly competitive solutions that yield on average 96 % of the value of an optimal solution. Y1 - 2021 ER - TY - INPR A1 - Klimm, Max A1 - Pfetsch, Marc A1 - Raber, Rico A1 - Skutella, Martin T1 - Reduction of Potential-Based Flow Networks N2 - We consider potential-based flow networks with terminal nodes at which flow can enter or leave the network and physical properties such as voltages or pressures are measured and controlled. We study conditions under which such a network can be reduced to a smaller, equivalent network with the same behavior at the terminal nodes. Potential-based flow networks are widely used to model infrastructure networks such as electricity, gas, or water networks. In contrast to Kron's reduction for electrical networks, we prove that, in general, potential-based flow networks with at least three terminals cannot be reduced to smaller networks whose size only depends on the number of terminals. On the other hand, we show that it is possible to represent a special class of potential-based flow networks by a complete graph on the terminals, and we establish a characterization of networks that can be reduced to a path network. Our results build on fundamental properties of effective resistances proved in this paper, including explicit formulae for their dependence on edge resistances of the network and their metric properties. Y1 - 2021 ER - TY - INPR 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. KW - Keller-Segel KW - chemotaxis KW - nonlinear diffusion KW - discontinuous Galerkin scheme KW - a posteriori error analysis Y1 - 2023 ER - TY - INPR 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. Y1 - 2023 ER - TY - JOUR 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. Y1 - U6 - https://doi.org/https://doi.org/10.3390/a15090310 VL - Algorithms IS - Vol. 15 ER - TY - JOUR 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. Y1 - VL - Journal of Computational Physics IS - Vol. 481 SP - 112041 ER - TY - JOUR 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. Y1 - U6 - 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 ER - TY - INPR 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. Y1 - 2023 VL - http://arxiv.org/abs/2303.18180 ER - TY - INPR 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. Y1 - ER - TY - INPR A1 - Schäfer Aguilar, Paloma A1 - Ulbrich, Stefan T1 - Convergence of numerical adjoint schemes arising from optimal boundary control problems of hyperbolic conservation laws N2 - We study the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal boundary control problems governed by entropy solutions of conservation laws. As boundary control we consider piecewise continuously differentiable controls with possible discontinuities at switching times, where the smooth parts as well as the switching times serve as controls. The derivative of tracking-type objective functionals with respect to the smooth controls and the switching times can then be represented by an adjoint-based formula. The main difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and boundary conditions that lead in general to discontinuous adjoints. Moreover, the solution of the adjoint equation is non-unique and the so-called reversible solution leads to the correct adjoint-based derivative representation. We study discrete adjoint schemes of monotone difference schemes in conservation form such as Engquist-Osher or Lax-Friedrichs scheme. We also allow that the state is computed by another numerical scheme satisfying certain convergence properties. We proof convergence results of the discrete adjoint to the reversible solution. Y1 - 2021 ER - TY - JOUR A1 - Giesselmann, Jan A1 - Gugat, Martin A1 - Kunkel, Teresa T1 - Observer-based data assimilation for barotropic gas transport using distributed measurements N2 - We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks. Y1 - 2023 U6 - https://doi.org/10.4310/CMS.240918203214 VL - 22 SP - 2271 EP - 2309 PB - Communications in Mathematical Sciences 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 - 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. KW - Probabilistic Robustness KW - Gas Network Control KW - Probabilistic Constrained Optimization KW - Stochastic Collocation KW - Kernel Density Estimation Y1 - 2025 ER - TY - INPR 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. KW - potential based flows KW - topology optimization KW - MINLP 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 - 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 - 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 - 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 - 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 -