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- 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.
- ItemUncertainty Quantification in Image Segmentation Using the Ambrosio–Tortorelli Approximation of the Mumford–Shah Energy(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2021) 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.
- ItemA function space framework for structural total variation regularization with applications in inverse problems(Bristol [u.a.] : Inst., 2018) Hintermüller, Michael; Holler, Martin; Papafitsoros, KostasIn this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable TV type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted TV for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.
- ItemVariable Step Mollifiers and Applications(Berlin ; Heidelberg : Springer, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections. © 2020, The Author(s).
- ItemOptimization with learning-informed differential equation constraints and its applications(Les Ulis : EDP Sciences, 2022) 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.
- ItemOptimality Conditions and Moreau-Yosida Regularization for Almost Sure State Constraints(Paris : EDP Sciences, 2022) Geiersbach, Caroline; Hintermüller, MichaelWe 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.
- ItemA generalized $Gamma$-convergence concept for a type of equilibrium problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hintermüller, Michael; Stengl, Steven-MarianA novel generalization of Γ-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ-convergence notion for equilibrium problems, generalizing the existing one from optimization, is introduced and discussed. The work ends with its application to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities.
- ItemA class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Dong, Guozhi; Hintermüller, Michael; Zhang, YeIn this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.
- ItemSimulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Gräßle, Carmen; Hintermüller, Michael; Hinze, Michael; Keil, TobiasWe are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.
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