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- ItemConstrained exact boundary controllability of a semilinear model for pipeline gas flow(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Gugat, Martin; Habermann, Jens; Hintermüller, Michael; Huber, OlivierWhile 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.
- ItemOn the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games (changed title: Vector-valued convexity of solution operators with application to optimal control problems)(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Stengl, Steven-MarianGeneralized Nash equilibrium problems in function spaces involving PDEs are considered. One of the central issues arising in this context is the question of existence, which requires the topological characterization of the set of minimizers for each player of the associated Nash game. In this paper, we propose conditions on the operator and the functional that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize results of convex analysis to derive optimality systems also for non-smooth operators. Our theoretical findings are illustrated by examples.
- ItemUncertainty quantification in image segmentation using the Ambrosio--Tortorelli approximation of the Mumford--Shah energy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Stengl, Steven-Marian; Surowiec, Thomas M.The quantification of uncertainties in image segmentation based on the Mumford-Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio-Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.
- ItemOptimal control and directional differentiability for elliptic quasi-variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
- ItemOn the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.
- ItemOptimization with learning-informed differential equation constraints and its applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, KostasInspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
- ItemFirst-order conditions for the optimal control of learning-informed nonsmooth PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas; Völkner, KathrinIn this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability.
- ItemDualization and automatic distributed parameter selection of total generalized variation via bilevel optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.; Sun, HongpengTotal Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.
- ItemDifferentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel(New York, NY : Springer, 2023) Hintermüller, Michael; Kröner, AxelIn this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.
- ItemExistence, iteration procedures and directional differentiability for parabolic QVIs(Berlin ; Heidelberg : Springer, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.