@unpublished{GugatHabermannHintermuelleretal.2021, author = {Gugat, Martin and Habermann, Jens and Hinterm{\"u}ller, Michael and Huber, Olivier}, title = {Constrained exact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exact boundary controllability of the system with the same pressure and Mach number constraints.}, language = {en} } @unpublished{DomschkeHillerLangetal.2021, author = {Domschke, Pia and Hiller, Benjamin and Lang, Jens and Mehrmann, Volker and Morandin, Riccardo and Tischendorf, Caren},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} }