@unpublished{GugatHabermannHintermuelleretal.2021, author = {Gugat, Martin and Habermann, Jens and Hinterm{\"u}ller, Michael and Huber, Olivier}, title = {Constrained exact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} } @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{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} } @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} } @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} } @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} } @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} } @unpublished{GrimmHintermuellerHuberetal., author = {Grimm, Veronika and Hinterm{\"u}ller, Michael and Huber, Olivier and Schewe, Lars and Schmidt, Martin and Z{\"o}ttl, Gregor}, title = {A PDE-Constrained Generalized Nash Equilibrium Approach for Modeling Gas Markets with Transport}, abstract = {We investigate a class of generalized Nash equilibrium problems (GNEPs) in which the objectives of the individuals are interdependent and the shared constraint consists of a system of partial differential equations. This setup is motivated by the modeling of strategic interactions of competing firms, which explicitly take into account the dynamics of transporting a commodity, such as natural gas, through a network. We establish the existence of a variational equilibrium of the GNEP. In the case of symmetric firms, we identify an equivalent optimization problem. We use this model to numerically explore the impact of linepacking, that is the use of the network as a temporary storage device. In particular, we study the firms' decisions under various linepacking abilities and analyze which market participants benefit from it.}, language = {en} }