TY - JOUR 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. KW - Turnpike Property KW - Wave Equation KW - Optimal Control KW - Spectral Method KW - Network Optimum Design Y1 - 2024 VL - J. Geom. Anal. IS - 34 ER - TY - INPR 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. KW - Euler Equations KW - quasilinear hyperbolic system KW - gas transport modeling KW - hydrogen blending Y1 - 2024 ER - TY - INPR 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. KW - Gas Networks KW - Compressor Control KW - Weber Problem KW - Optimal Location KW - Uncertain Boundary Data Y1 - 2024 ER - TY - INPR 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. KW - Probabilistic Constrained Optimization KW - Stochastic Optimization KW - Chance Constraints KW - Kernel Density Estimation KW - Convergence Analysis Y1 - 2024 ER - TY - INPR A1 - Hante, Falk M. A1 - Schmidt, Martin A1 - Topalovic, Antonia T1 - Stabilizing GNEP-Based Model Predictive Control: 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 if stabilizing end 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 end 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 ER - TY - INPR A1 - Kuchlbauer, Martina T1 - Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems N2 - We consider nonlinear robust optimization problems with mixed-integer decisions as well as nonconvexities. In detail, we consider cases where objective and constraint functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex. We propose an algorithm for such robust optimization problems that does not require a certain structure of the adversarial problem but only requires that approximate worst cases are available. As a result, our algorithm finds a robust optimal solution up to a tolerance. Our method integrates a bundle method into an outer approximation approach where the bundle method is used for the arising continuous subproblems. We rely on methods from the literature, namely a bundle method for nonlinear and nonconvex robust optimization problems and outer approximation approaches for quasiconvex settings. Our contribution is to combine them to one convergent robust optimization method that can cope with inexactness of worst-case evaluations. Further, we propose the gas transport under uncertainties as a relevant application and demonstrate that generalized convexity is fulfilled for a type of a network structure. Y1 - ER - TY - INPR 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. KW - Robust Optimization KW - Nonlinear Flows KW - Potential-based Networks KW - Demand Uncertainties KW - Mixed-integer Nonlinear Optimization Y1 - 2024 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 - 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 - 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 - 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 - 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 - 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 - 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. Y1 - 2024 ER - TY - INPR 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. Y1 - 2023 U6 - https://doi.org/https://doi.org/10.5802/smai-jcm.93 VL - 9 ER - TY - INPR 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. KW - Keller-Segel KW - chemotaxis; KW - nonlinear diffusion KW - finite volume scheme KW - a posteriori error analysis Y1 - 2023 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 - 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. KW - Conservation laws KW - entropy conditions KW - entropy solutions KW - shocks, KW - a posteriori error estimates Y1 - 2023 ER - TY - INPR 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. KW - gas transport KW - doubly nonlinear parabolic problems KW - relative entropy estimates KW - exponential stability KW - structure preserving discretization Y1 - 2023 ER - TY - INPR 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. Y1 - 2023 ER -