@unpublished{GugatHabermannHintermuelleretal.2021, author = {Gugat, Martin and Habermann, Jens and Hinterm{\"u}ller, Michael and Huber, Olivier}, title = {Constrained exact boundary controllability of a semilinear model for pipeline gas flow}, year = {2021}, abstract = {While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.}, language = {en} } @unpublished{DomschkeHillerLangetal.2021, author = {Domschke, Pia and Hiller, Benjamin and Lang, Jens and Mehrmann, Volker and Morandin, Riccardo and Tischendorf, Caren}, title = {Gas Network Modeling: An Overview}, pages = {53}, year = {2021}, abstract = {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. The idea of a model catalog came to us in the context of the application for the CRC/Transregio 154 ``Mathematical modeling, simulation and optimization using the example of gas networks''. The present English translation is an extension from [P. Domschke, B. Hiller, J. Lang, and C. Tischendorf. Modellierung von Gasnetzwerken: Eine {\"U}bersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support.}, language = {en} } @article{GugatHanteJin2020, author = {Gugat, Martin and Hante, Falk and Jin, Li}, title = {Closed loop control of gas flow in a pipe: Stability for a transient model}, series = {at - Automatisierungstechnik}, journal = {at - Automatisierungstechnik}, year = {2020}, abstract = {This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.}, language = {en} } @article{HajianHintermuellerUlbrich2017, author = {Hajian, Soheil and Hinterm{\"u}ller, Michael and Ulbrich, Stefan}, title = {Total variation diminishing schemes in optimal control of scalar conservation laws}, series = {IMA Journal of Numerical Analysis}, volume = {39}, journal = {IMA Journal of Numerical Analysis}, doi = {10.20347/WIAS.PREPRINT.2383}, pages = {105 -- 140}, year = {2017}, abstract = {In this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented.}, language = {en} } @article{PapafitsorosHintermuellerRautenberg2016, author = {Papafitsoros, Konstantinos and Hinterm{\"u}ller, Michael and Rautenberg, Carlos}, title = {Analytical aspects of spatially adapted total variation regularisation}, series = {Journal of Mathematical Analysis and Applications}, volume = {454}, journal = {Journal of Mathematical Analysis and Applications}, pages = {891 -- 935}, year = {2016}, abstract = {In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions.}, language = {en} } @article{KeilHintermuellerWegner, author = {Keil, Tobias and Hinterm{\"u}ller, Michael and Wegner, Donat}, title = {Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities}, series = {SIAM Journal on Control and Optimization}, volume = {55}, journal = {SIAM Journal on Control and Optimization}, pages = {1954 -- 1989}, abstract = {This paper is concerned with the distributed optimal control of a time-discrete Cahn- Hilliard/Navier-Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier-Stokes equation. By proposing a suitable time- discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau-Yosida type approximations of the double-obstacle potential. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-stationarity which is a special notion of stationarity for MPECs.}, language = {en} } @article{HintermuellerRautenbergMohammadietal.2016, author = {Hinterm{\"u}ller, Michael and Rautenberg, Carlos and Mohammadi, Masoumeh and Kanitsar, Martin}, title = {Optimal sensor placement: A robust approach}, series = {SIAM Journal on Control and Optimization}, volume = {55}, journal = {SIAM Journal on Control and Optimization}, pages = {3609 -- 3639}, year = {2016}, abstract = {We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.}, language = {en} } @article{GonzalezGrandonHeitschHenrion2017, author = {Gonzalez Grandon, Tatiana and Heitsch, Holger and Henrion, Rene}, title = {A joint model of probabilistic/robust constraints for gas transport management in stationary networks}, series = {Computational Management Science}, volume = {14}, journal = {Computational Management Science}, doi = {10.1007/s10287-017-0284-7}, pages = {443 -- 460}, year = {2017}, abstract = {We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.}, language = {en} } @unpublished{HuckTischendorf, author = {Huck, Christoph and Tischendorf, Caren}, title = {Perturbation analysis of hyperbolic PDAEs describing flow networks}, language = {en} } @incollection{BennerGrundelHimpeetal., author = {Benner, Peter and Grundel, Sarah and Himpe, Christian and Huck, Christoph and Streubel, Tom and Tischendorf, Caren}, title = {Gas Network Benchmark Models}, series = {Applications of Differential-Algebraic Equations: Examples and Benchmarks}, volume = {DAE-FORUM}, booktitle = {Applications of Differential-Algebraic Equations: Examples and Benchmarks}, publisher = {Springer}, address = {Cham}, doi = {10.1007/11221_2018_5}, pages = {171 -- 197}, abstract = {The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models.}, language = {en} } @inproceedings{HuckTischendorf, author = {Huck, Christoph and Tischendorf, Caren}, title = {Transient Modeling and Simulation of Gas Pipe Networks with Characteristic Diagram Models for Compressors}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {17}, booktitle = {Proceedings in Applied Mathematics and Mechanics}, number = {1}, doi = {10.1002/pamm.201710322}, pages = {707 -- 708}, abstract = {One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4, 7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1-3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system.}, language = {en} } @article{HintermuellerStrogies2017, author = {Hinterm{\"u}ller, Michael and Strogies, Nikolai}, title = {On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws}, series = {Numerical Analysis and Optimization}, journal = {Numerical Analysis and Optimization}, doi = {10.20347/WIAS.PREPRINT.2442}, pages = {119 -- 154}, year = {2017}, abstract = {Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.}, language = {en} } @article{HintermuellerRautenbergStrogies2017, author = {Hinterm{\"u}ller, Michael and Rautenberg, Carlos and Strogies, Nikolai}, title = {Dissipative and Non-dissipative Evolutionary Quasi-variational Inequalities with Gradient Constraints}, series = {Set-Valued and Variational Analysis}, volume = {27}, journal = {Set-Valued and Variational Analysis}, pages = {433 -- 468}, year = {2017}, abstract = {Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.}, language = {en} } @unpublished{DomschkeHillerLangetal.2017, author = {Domschke, Pia and Hiller, Benjamin and Lang, Jens and Tischendorf, Caren}, title = {Modellierung von Gasnetzwerken: Eine {\"U}bersicht}, volume = {2717}, pages = {33}, year = {2017}, abstract = {Mit dieser {\"U}bersicht wollen wir eine Zusammenstellung von unterschiedlichen Modellen zur Beschreibung des Gasflusses in Netzwerken bereitstellen, um den Einstieg in das Thema zu erleichtern. Besonderes Augenmerk wird dabei auf die der Modellierung inneliegende hierarchische Struktur und die detaillierte Beschreibung einzelner Bauteile wie Ventile und Kompressoren gelegt. Daneben finden sich ebenfalls Netzmodellklassen, die auf rein algebraische Relationen aufbauen. Am Ende geben wir einen kurzen {\"U}berblick {\"u}ber grundlegende numerische Verfahren und Konzepte zur Behandlung von hyperbolischen Bilanzgleichungen. Wir erheben keinen Anspruch auf Vollst{\"a}ndigkeit und verweisen an vielen Stellen auf die bestehende Literatur. Die Idee eines Modellkataloges ist uns im Rahmen der Antragstellung zum SFB/Transregio 154 „Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken" gekommen. Wir m{\"o}chten an dieser Stelle die F{\"o}rderung durch die DFG dankend erw{\"a}hnen.}, language = {de} } @article{LeoeveyHeitschRoemisch2016, author = {Le{\"o}vey, Hernan and Heitsch, Holger and R{\"o}misch, Werner}, title = {Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?}, series = {Computational Optimization and Applications}, volume = {65}, journal = {Computational Optimization and Applications}, number = {3}, doi = {10.1007/s10589-016-9843-z}, pages = {567 -- 603}, year = {2016}, abstract = {Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence O(n-1+δ ) with δ ∈ (0, 12 ] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.}, language = {en} } @article{HintermuellerSurowiec2015, author = {Hinterm{\"u}ller, Michael and Surowiec, Thomas}, title = {A Bundle-Free Implicit Programming Approach for a Class of MPECs in Function Space}, series = {Mathematical Programming}, volume = {160}, journal = {Mathematical Programming}, number = {1-2}, doi = {10.1007/s10107-016-0983-9}, pages = {271 -- 305}, year = {2015}, abstract = {Using a standard first-order optimality condition for nonsmooth optimization prob- lems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.}, language = {en} } @article{HintermuellerSurowiec2016, author = {Hinterm{\"u}ller, Michael and Surowiec, Thomas}, title = {On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind}, series = {Set-Valued and Variational Analysis}, volume = {26}, journal = {Set-Valued and Variational Analysis}, doi = {10.1007/s11228-017-0408-9}, pages = {631 -- 642}, year = {2016}, abstract = {The directional differentiability of the solution mapping for a class of variational inequali- ties of the second kind inspired by applications in fluid mechanics and moving free boundary problems is investigated. The result is particularly relevant for the model predictive control or optimal control of such variational inequalities in that it can be used to derive stationarity conditions and efficient numerical methods.}, language = {en} } @phdthesis{Huck2018, author = {Huck, Christoph}, title = {Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks}, doi = {10.18452/19596}, year = {2018}, abstract = {This thesis addresses several aspects regarding modelling, analysis and numerical simulation of gas networks. Hereby, our focus lies on (partial) differential-algebraic equations, thus systems of partial and ordinary differential equations which are coupled by algebraic equations. These coupled systems allow an easy approach towards the modelling of dynamic structures on networks. Therefore, they are well suited for gas networks, which have gained a rise of attention in society, politics and science due to the focus towards renewable energies. We give an introduction towards gas network modelling that includes the most common elements that also appear in real gas networks and present two PDAE systems: One for pipe networks and one that includes additional elements like resistors and compressors. Furthermore, we investigate the impact of perturbations onto the pipe network PDAE, where we explicitly allow perturbations to affect the system in the differential as well as in the algebraic components. We conclude that the solution of the PDAE possesses stability properties. In addition, this thesis introduces a new spatial discretisation that is adapted to the net- work topology. This topology-adapted semi-discretisation results in a DAE which possesses the same perturbation behaviour as the space continuous PDAE. Furthermore, we present a topology based decoupling procedure that allows to reformulate the DAE as an ordinary differential equation (ODE), which represents the inherent dynamics of the DAE system. This ODE, together with a decoupled set of algebraic equations, can be derived from the topology and element information directly. We conclude by demonstrating the established results for several benchmark networks. This includes a comparison of numerical solutions for the decoupled ODE and the DAE system. In addition we present the advantages of the topology-adapted spatial discretisation over existing well established methods.}, language = {en} } @unpublished{ShyshkanovaWalther2022, author = {Shyshkanova, Ganna and Walther, Andrea}, title = {Contact Pressure over Doubly Connected Rectangular Domains and Punch Shape Optimization}, year = {2022}, abstract = {Contact problems arise in a variety of industrial processes, engineering and biomechanical systems. 3-D contact problem for a rigid punch with a doubly connected base bounded by the lines close to rectangles is in consideration. An analytic-numerical technique is developed for its solving. The problem contains Fredholm integral equations of the first kind, which are transformed into the second kind by means of regularization. Using the simple layer potential expansion, the kernels of the integrals are presented in the form of expansions in the powers of the polar radius. The difference between the values of the desired function at different points and the subsequent interpolation of the terms are proposed to smooth the kernels and eliminate singularities. The integral equations are reduced to one-dimension and then solved using quadrature formulas. Subsequently a punch shape is taken as a desired function, and as a minimizing functional is considered the root-mean-square deviation of the pressure distribution arising under the punch from some optimal distribution. In this case, the values of the total forces and moments applied to the punch are assumed to be given, which leads to restrictions imposed on the distributions by the equilibrium conditions. The normal displacements are determined which arising under the action of the found contact pressure on the elastic half-space. The desired punch shape is found using the simple layer potential. A solution to the problem is obtained for the punch with the doubly connected base bounded by lines close to rectangles.}, language = {en} } @article{KreimeierKuchlbauerLiersetal.2022, author = {Kreimeier, Timo and Kuchlbauer, Martina and Liers, Frauke and Stingl, Michael and Walther, Andrea}, title = {Towards the Solution of Robust Gas Network Optimization Problems Using the Constrained Active Signature Method}, year = {2022}, abstract = {This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research.}, language = {en} } @unpublished{GahururuHintermuellerSurowiec2021, author = {Gahururu, Deborah and Hinterm{\"u}ller, Michael and Surowiec, Thomas}, title = {Risk-Neutral PDE-Constrained Generalized Nash Equilibrium Problems}, year = {2021}, abstract = {A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player's actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples.}, language = {en} } @unpublished{GeiersbachHintermueller2021, author = {Geiersbach, Caroline and Hinterm{\"u}ller, Michael}, title = {Optimality conditions and Moreau-Yosida regularization for almost sure state constraints}, year = {2021}, abstract = {We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau-Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.}, language = {en} } @unpublished{Hante, author = {Hante, Falk}, title = {Relaxation methods for optimal switching control of PDE-dynamical systems}, abstract = {The chapter reviews certain computational approaches to solve optimal control problems for evolution-type partial differential equations, where some control functions are limited to switching. The mechanism that enforces switching is modeled as integer restrictions. This brings a combinatorial aspect into the apart from switching already computationally very demanding optimization problems. Recently, great advances have been made to tackle such problems rigorously using relaxation and combinatorial integral approximation. An overview of these methods and the known theoretical results concerning convergence and error estimates are provided in a consistent manner. Further, we point to applications with benchmark character as well as to open problems.}, language = {en} } @unpublished{KreimeierSauterStreubeletal.2021, author = {Kreimeier, Timo and Sauter, Henning and Streubel, Tom and Tischendorf, Caren and Walther, Andrea}, title = {Solving Least-Squares Collocated Differential Algebraic Equations by Successive Abs-Linear Minimization - A Case Study on Gas Network Simulation}, year = {2021}, abstract = {This paper studies the numerical simulation of gas networks with regulating elements using differential algebraic equations (DAEs) in combination with least-squares collocation. In contrast to classical collocation methods, more collocation points than degrees of freedom for the collocation polynomials are used. Recently, it has been shown that such a least-squares collocation has a regularizing effect for DAEs, in particular for DAEs with higher index. In each time step of the numerical integration, one has to solve a system of nonlinear equations that is nonsmooth due to the regulating elements in the gas networks. We consider four solvers one of which explicitly exploits the inherent nonsmooth nature. Numerical results are given for three different test cases with increasing complexity illustrating the feasibility of the proposed approach to approximate a solution of the DAE and the advantageous performance of the nonsmooth solver that is based on the concept of abs-linearization.}, language = {en} } @unpublished{SauterTischendorf2021, author = {Sauter, Henning and Tischendorf, Caren}, title = {Optimal control of gas network DAEs}, year = {2021}, language = {en} } @unpublished{SauterTischendorf2021, author = {Sauter, Henning and Tischendorf, Caren}, title = {Least squares collocation for the simulation of gas network DAEs}, year = {2021}, language = {en} } @unpublished{HanteKrugSchmidt2021, author = {Hante, Falk and Krug, Richard and Schmidt, Martin}, title = {Time-Domain Decomposition for Mixed-Integer Optimal Control Problems}, pages = {32}, year = {2021}, abstract = {We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems.}, language = {en} } @unpublished{HuckTischendorf, author = {Huck, Christoph and Tischendorf, Caren}, title = {Topology motivated discretization of hyperbolic PDAEs describing flow networks}, language = {en} } @unpublished{SchmidtHillerKochetal.2019, author = {Schmidt, Martin and Hiller, Benjamin and Koch, Thorsten and Pfetsch, Marc and Geißler, Bj{\"o}rn and Henrion, Ren{\´e} and Joormann, Imke and Martin, Alexander and Morsi, Antonio and R{\"o}misch, Werner and Schewe, Lars and Schultz, R{\"u}diger and Steinbach, Marc C.}, title = {Capacity Evaluation for Large-Scale Gas Networks}, pages = {8}, year = {2019}, abstract = {Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20\% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance. As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty. The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2019, author = {Klimm, Max and Pfetsch, Marc E. and Raber, Rico and Skutella, Martin}, title = {Packing under Convex Quadratic Constraints}, year = {2019}, abstract = {We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding method leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three algorithms for problem instances arising in the context of real-world gas transport networks.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2019, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {On the Robustness of Potential-Based Flow Networks}, year = {2019}, abstract = {Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., electricity, gas, or water networks, where the flow along each edge is controlled via the (difference of) potentials at its end nodes. A potential-based flow network is robust if the maximal difference of node potentials needed to satisfy a set of demands cannot increase if demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual amounts sent later on. Here node potentials correspond to physical quantities such as the pressures or the voltages and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of such robust networks for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness.}, language = {en} } @article{AdelhuetteAssmannGonzalezGrandonetal.2017, author = {Adelh{\"u}tte, Dennis and Aßmann, Denis and Gonz{\`a}lez Grand{\`o}n, Tatiana and Gugat, Martin and Heitsch, Holger and Liers, Frauke and Henrion, Ren{\´e} and Nitsche, Sabrina and Schultz, R{\"u}diger and Stingl, Michael and Wintergerst, David}, title = {Joint model of probabilistic/robust (probust) constraints applied to gas network optimization}, doi = {10.1007/s10013-020-00434-y}, year = {2017}, abstract = {Optimization tasks under uncertain conditions abound in many real-life applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound, with high probability.}, language = {en} } @article{HintermuellerStrogies2017, author = {Hinterm{\"u}ller, Michael and Strogies, Nikolai}, title = {Identification of the friction function in a semilinear system for gas transport through a network}, series = {Optimization Methods and Software}, volume = {35}, journal = {Optimization Methods and Software}, pages = {576 -- 617}, year = {2017}, abstract = {An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions is established. Finally, numerical experiments validate the theoretical findings.}, language = {en} } @unpublished{BrodskyiHanteSeidel, author = {Brodskyi, Yan and Hante, Falk and Seidel, Arno}, title = {Stabilization of Partial Differential Equations by Sequential Action Control}, abstract = {We extend the framework of sequential action control to systems of partial differential equations which can be posed as abstract linear control problems in a Hilbert space. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for quadratic stage costs. We apply this theory prototypically to an unstable heat equation and verify the results numerically.}, language = {en} } @article{GoettlichHantePotschkaetal., author = {G{\"o}ttlich, Simone and Hante, Falk and Potschka, Andreas and Schewe, Lars}, title = {Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints}, series = {Mathematical Programming}, journal = {Mathematical Programming}, abstract = {We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.}, language = {en} } @article{HajianHintermuellerSchillingsetal., author = {Hajian, Soheil and Hinterm{\"u}ller, Michael and Schillings, Claudia and Strogies, Nikolai}, title = {A Bayesian approach to parameter identification in gas networks}, series = {Control and Cybernetics}, volume = {48}, journal = {Control and Cybernetics}, pages = {377 -- 402}, abstract = {The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.}, language = {en} } @article{HankeMaerzTischendorf, author = {Hanke, Michael and M{\"a}rz, Roswitha and Tischendorf, Caren}, title = {Least-Squares Collocation for Higher-Index Linear Differential-Algebraic Equations: Estimating the Instability Threshold}, series = {Math. Comp.}, volume = {88}, journal = {Math. Comp.}, doi = {10.1090/mcom/3393}, pages = {1647 -- 1683}, abstract = {Differential-algebraic equations with higher index give rise to essentially ill-posed problems. The least-squares collocation by discretizing the pre-image space is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations and index-1 differential-algebraic equations. This approach has displayed excellent convergence properties in numerical experiments, however, theoretically, till now convergence could be established merely for regular linear differential-algebraic equations with constant coefficients. We present now an estimate of the instability threshold which serves as the basic key for proving convergence for general regular linear DAEs.}, language = {en} } @unpublished{HanteSchmidt2023, author = {Hante, Falk M. and Schmidt, Martin}, title = {Gas Transport Network Optimization: Mixed-Integer Nonlinear Models}, pages = {8}, year = {2023}, abstract = {Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature.}, language = {en} } @unpublished{KreimeierPokuttaWaltheretal.2022, author = {Kreimeier, Timo and Pokutta, Sebastian and Walther, Andrea and Woodstock, Zev}, title = {On a Frank-Wolfe Approach for Abs-smooth Functions}, year = {2022}, abstract = {We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our nonsmooth nonconvex problem setting is motivated by machine learning, since the broad class of abs-smooth functions includes, for instance, the squared \$\ell_2\$-error of a neural network with ReLU or hinge Loss activation. To overcome the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is identical to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.}, language = {en} } @article{ShyshkanovaZaytsevaZhushmanetal.2023, author = {Shyshkanova, Ganna and Zaytseva, Tetyana and Zhushman, V and Levchenko, Ntaliia and Korotunova, Olena}, title = {Solving three-dimensional contact problems for foundation design in green building}, doi = {10.1088/1742-6596/2609/1/012001}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{KannanKreimeierWalther2023, author = {Kannan, Aswin and Kreimeier, Timo and Walther, Andrea}, title = {On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods}, year = {2023}, language = {en} } @unpublished{HanteKuchler, author = {Hante, Falk and Kuchler, Christian}, title = {Indirect methods for optimal control of hybrid PDE-dynamical / switching systems using relaxation}, abstract = {We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear evolution-type PDEs including a state-dependent control-regime switching mechanism. We introduce an equivalent mixed-integer formulation featuring vanishing constraints arising by methods of disjunctive programming. We embed the problem into the class of equilibrium constraints by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding 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 of the vanishing constraints. After the theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on our approach.}, language = {en} } @unpublished{HanteSchmidt2023, author = {Hante, Falk M. and Schmidt, Martin}, title = {Gas Transport Network Optimization: PDE-Constrained Models}, pages = {8}, year = {2023}, abstract = {The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems.}, language = {en} } @unpublished{BongartiHintermueller2023, author = {Bongarti, Marcelo and Hinterm{\"u}ller,}, title = {Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{KreimeierWaltherGriewank2021, author = {Kreimeier, Timo and Walther, Andrea and Griewank, Andreas}, title = {An active signature method for constrained abs-linear minimization}, year = {2021}, abstract = {In this paper we consider the solution of optimization tasks with a piecewise linear objective function and piecewise linear constraints. First, we state optimality conditions for that class of problems using the abs-linearization approach and prove that they can be verified in polynomial time. Subsequently, we propose an algorithm called Constrained Active Signature Method that explicitly exploits the piecewise linear structure to solve such optimization problems. Convergence of the algorithm within a finite number of iterations is proven. Numerical results for various testcases including linear complementarity conditions and a bi-level problem illustrate the performance of the new algorithm.}, language = {en} } @unpublished{HanteKuchler2023, author = {Hante, Falk and Kuchler, Christian}, title = {An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs}, year = {2023}, abstract = {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.}, language = {en} } @article{ShyshkanovaWalther2023, author = {Shyshkanova, Ganna and Walther, Andrea}, title = {Optimization of a punch shape with a doubly connected contact domain}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{GraserKreimeierWalther2024, author = {Graser, Gertrud and Kreimeier, Timo and Walther, Andrea}, title = {Solving Linear Generalized Nash Games Using an Active Signature Method}, year = {2024}, abstract = {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.}, language = {en} } @unpublished{DomschkeGiesselmannLangetal.2023, author = {Domschke, Pia and Giesselmann, Jan and Lang, Jens and Breiten, Tobias and Mehrmann, Volker and Morandin, Riccardo and Hiller, Benjamin and Tischendorf, Caren}, title = {Gas Network Modeling: An Overview (Extended English Version)}, pages = {54}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{HanteSchmidtTopalovic2024, author = {Hante, Falk M. and Schmidt, Martin and Topalovic, Antonia}, title = {Stabilizing GNEP-Based Model Predictive Control: Quasi-GNEPs and End Constraints}, pages = {26}, year = {2024}, abstract = {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.}, language = {en} }