Nichtglatte Variationsprobleme und Operatorgleichungen

Publikationen

Artikel in Referierten Journalen

• M. Bongarti, M. Hintermüller, Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 89 (2024), pp. 36/1--36/48, DOI 10.1007/s00245-023-10088-0 .
  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.

• G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, GAMM-Mitteilungen, pp. e202470014/1-- e202470014/35, DOI 10.1002/gamm.202470014 .
  Abstract
  In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

• G. Dong, M. Hintermüller, K. Papafitsoros, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Journal on Optimization, 34 (2024), pp. 2314--2349, DOI 10.1137/22M1534420 .
  Abstract
  We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.

• T. Gutleb, S. Olver, I. Papadopoulos, R. Slevinsky, Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli, SIAM Journal on Scientific Computing, 46 (2024), pp. A3448--A3476, DOI 10.1137/23M160846X .

• S. Olver, I. Papadopoulos, A sparse spectral method for fractional differential equations in one--spatial dimension, Advances in Computational Mathematics, 50 (2024), pp. 1--45, DOI 10.1007/s10444-024-10164-1 .

• C. Geiersbach, T. Suchan, K. Welker, Stochastic augmented Lagrangian method in Riemannian shape manifolds, Journal of Optimization Theory and Applications, published online on 21.08.2024, DOI 10.1007/s10957-024-02488-1 .
  Abstract
  In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically.

• C. Geiersbach, R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Mathematics of Operations Research, published online on 15.07.2024, DOI 10.1287/moor.2023.0177 .
  Abstract
  In this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.

• M. Hintermüller, S.-M. Stengl, A generalized $Gamma$-convergence concept for a type of equilibrium problems, Journal of Nonlinear Science, 34 (2024), pp. 83/1--83/28, DOI 10.1007/s00332-024-10059-x .
  Abstract
  A 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.

• M. Hintermüller, Th.M. Surowiec, M. Theiss, On a differential generalized Nash equilibrium problem with mean field interaction, SIAM Journal on Optimization, 34 (2024), pp. 2821--2855, DOI 10.1137/22M1489952 .

• I. Papadopoulos, Numerical analysis of the SIMP model for the topology optimization problem of minimizing compliance in linear elasticity, Numerische Mathematik, published online on 19.11.2024, DOI 10.1007/s00211-024-01438-3 .

Preprints, Reports, Technical Reports

• A. Alphonse, G. Wachsmuth, Subdifferentials and penalty approximations of the obstacle problem, Preprint no. 3159, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3159 .
  Abstract, PDF (331 kByte)
  We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalised problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving for example the positive part function max(0, ·). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.

• A. Alphonse, M. Hintermüller, A. Kister, Ch.H. Lun, C. Sirotenko, A neural network approach to learning solutions of a class of elliptic variational inequalities, Preprint no. 3152, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3152 .
  Abstract, PDF (21 MByte)
  We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.

• G. Dong, M. Hintermüller, C. Sirotenko, Dictionary learning based regularization in quantitative MRI: A nested alternating optimization framework, Preprint no. 3135, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3135 .
  Abstract, PDF (5706 kByte)
  In this article we propose a novel regularization method for a class of nonlinear inverse problems that is inspired by an application in quantitative magnetic resonance imaging (MRI). It is a special instance of a general dynamical image reconstruction problem with an underlying time discrete physical model. Our regularization strategy is based on dictionary learning, a method that has been proven to be effective in classical MRI. To address the resulting non-convex and non-smooth optimization problem, we alternate between updating the physical parameters of interest via a Levenberg-Marquardt approach and performing several iterations of a dictionary learning algorithm. This process falls under the category of nested alternating optimization schemes. We develop a general such algorithmic framework, integrated with the Levenberg-Marquardt method, of which the convergence theory is not directly available from the literature. Global sub-linear and local strong linear convergence in infinite dimensions under certain regularity conditions for the sub-differentials are investigated based on the Kurdyka?Lojasiewicz inequality. Eventually, numerical experiments demonstrate the practical potential and unresolved challenges of the method.

• A. Alphonse, C. Christof, M. Hintermüller, I. Papadopoulos, A globalized inexact semismooth Newton method for nonsmooth fixed--point equations involving variational inequalities, Preprint no. 3132, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3132 .
  Abstract, PDF (23 MByte)
  We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q-superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm.

• A. Alphonse, D. Caetano, Ch.M. Elliott, Ch. Venkataraman, Free boundary limits of coupled bulk--surface models for receptor--ligand interactions on evolving domains, Preprint no. 3122, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3122 .
  Abstract, PDF (5947 kByte)
  We derive various novel free boundary problems as limits of a coupled bulk-surface reaction-diffusion system modelling ligand-receptor dynamics on evolving domains. These limiting free boundary problems may be formulated as Stefan-type problems on an evolving hypersurface. Our results are new even in the setting where there is no domain evolution. The models are of particular relevance to a number of applications in cell biology. The analysis utilises L^∞-estimates in the manner of De Giorgi iterations and other technical tools, all in an evolving setting. We also report on numerical simulations.

• M. Dambrine, C. Geiersbach, H. Harbrecht, Two--norm discrepancy and convergence of the stochastic gradient method with application to shape optimization, Preprint no. 3121, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3121 .
  Abstract, PDF (447 kByte)
  The present article is dedicated to proving convergence of the stochastic gradient method in case of random shape optimization problems. To that end, we consider Bernoulli's exterior free boundary problem with a random interior boundary. We recast this problem into a shape optimization problem by means of the minimization of the expected Dirichlet energy. By restricting ourselves to the class of convex, sufficiently smooth domains of bounded curvature, the shape optimization problem becomes strongly convex with respect to an appropriate norm. Since this norm is weaker than the differentiability norm, we are confronted with the so-called two-norm discrepancy, a well-known phenomenon from optimal control. We therefore need to adapt the convergence theory of the stochastic gradient method to this specific setting correspondingly. The theoretical findings are supported and validated by numerical experiments.

• G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, Preprint no. 3105, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3105 .
  Abstract, PDF (7590 kByte)
  In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

• A. Alphonse, M. Hintermüller, C.N. Rautenberg, G. Wachsmuth, Minimal and maximal solution maps of elliptic QVIs: Penalisation, Lipschitz stability, differentiability and optimal control, Preprint no. 3093, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3093 .
  Abstract, PDF (501 kByte)
  Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau?Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.

Vorträge, Poster

• Q. Wang, Robust multilevel training of artificial neural networks, Math+ Days, Berlin, October 18, 2024.

• M. Fröhlich, Quantum noise characterization with a tensor network quantum jump method, Workshop on Tensor Methods for Quantum Simulation 2024, June 3 - 7, 2024, Zuse-Institut Berlin (ZIB), June 7, 2024.

• M. Fröhlich, Tensor network quantum jump method for quantum noise characterization, Einstein Research Unit, 4th Research Update Meeting, June 13, 2024, Magnus Haus, Berlin.

• D. Korolev, A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem, Leibniz MMS Days 2024, Parallel Session ``Computational Material Science'', April 10 - 12, 2024, Leibniz-Institut für Verbundwerkstoffe (IVW), Kaiserslautern, April 11, 2024.

• I. Papadopoulos, A frame approach for equations involving the fractional Laplacian, Singular and oscillatory integration, June 24 - 26, 2024, University College London, Department of Mathematics, UK, June 25, 2024.

• I. Papadopoulos, A semismooth Newton method for obstacle--type quasivariational inequalities, Firedrake 2024, September 16 - 18, 2024, University of Oxford, UK, September 18, 2024.

• F. Sauer, Equilibria for Distributed Multi-Modal Energy Systems under Uncertainty, Math+ Days, Berlin, October 18, 2024.

• M. Brokate, Derivatives of rate-independent evolutions, IFIP TC7 System Modeling and Optimization, August 12 - 16, 2024, Universität Hamburg, August 15, 2024.

• M. Brokate, Differential sensitivity in rate independent problems, Control of state constrained dynamical systems, September 25 - 29, 2024, Universita di Padova, Italy, September 26, 2024.

• C. Geiersbach, Basics of random algorithms, TRR 154 summer school on ``Optimization, Uncertainty and AI'', August 7 - 8, 2024, Universität Hamburg, August 8, 2024.

• C. Geiersbach, Numerical solution of an optimal control problem with probabilistic or almost sure state constraints, MS63 2024 SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS63: ``Efficient Solution Schemes for Optimization of Complex Systems Under Uncertainty'', February 27 - March 1, 2024, Trieste, Italy, February 28, 2024.

• C. Geiersbach, Optimality conditions with probabilistic state constraints, ISMP 2024 -- 25th International Symposium on Mathematical Programming, Session TB111 ``PDE--constrained optimization under uncertainty'', July 21 - 26, 2024, Montreal, Canada, July 23, 2024.

• C. Geiersbach, Optimization with probabilistic state constraints, Workshop ``Control and Optimization in the Age of Data'', September 18 - 20, 2024, Universität Bayreuth, September 19, 2024.

• C. Geiersbach, PDE-restringierte Optimierungsprobleme mit probabilistischen Zustandsschranken, Women in Optimization 2024, April 10 - 12, 2024, Friedrich-Alexander-Universität Erlangen (FAU), April 10, 2024.

• C. Geiersbach, Probabilistic state constraints for optimal control problems under uncertainty, VARANA 2024: Variational analysis and applications, September 1 - 7, 2024, International School of Mathematics ``Guido Stampacchia'', Erice, Italy, September 2, 2024.

• C. Geiersbach, Stochastic approximation for PDE-constrained optimization under uncertainty, Summer School on Numerical methods for random differential models, June 11 - 14, 2024, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, June 12, 2024.

• A. Alphonse, A quasi-variational contact problem arising in thermoelasticity, Workshop ``Interfaces, Free Boundaries and Geometric Partial Differential Equations'', February 12 - 16, 2024, Mathematisches Forschungsinstitut Oberwolfach, February 15, 2024.

• A. Alphonse, Risk--averse optimal control of elliptic variational inequalities, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 19.01 ``Various topics in Optimization of Differential Equations (1)'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 19, 2024.

• M. Hintermüller, A hybrid physics-informed neural network based multiscale solver as a PDE constrained optimization problem, ISMP 2024 -- 25th International Symposium on Mathematical Programming, Session TA90 ``Nonsmooth PDE Constrained Optimization'', July 21 - 26, 2024, Montreal, Canada, July 23, 2024.

• M. Hintermüller, A neural network approach to learning solutions of a class of elliptic variational inequalities, Nonsmooth Optimization and Variational Analysis, December 2 - 4, 2024, The Hong Kong Polytechnic University, China, December 4, 2024.

• M. Hintermüller, A neural network approach to learning solutions of a class of elliptic variational inequalities, Chinese University of Hong Kong, Department of Mathematics, China, December 12, 2024.

• M. Hintermüller, A neural network approach to learning solutions of a class of elliptic variational inequalities, colloquium talk, Hunan Normal University, Department of Mathematics, Changsha, China, December 10, 2024.

• M. Hintermüller, PDE-constrained optimization with learning-informed structures, Recent Advances in Scientific Computing and Inverse Problems, March 11 - 12, 2024, The Hong Kong Polytechnic University, China, March 11, 2024.

• M. Hintermüller, PINNs in multiscale materials as PDE-constrained optimization problem, Forum for Mathematical Optimization with PDEs, Central South University, Changsha, China, December 9, 2024.

• M. Hintermüller, QVIs: Semismooth Newton, optimal control and uncertainties, colloquium talk, Hunan Normal University, Department of Mathematics, Changsha, China, December 10, 2024.

• M. Hintermüller, QVIs: Semismooth Newton, optimal control, and uncertainties, RICAM Colloquium, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, June 27, 2024.

• M. Hintermüller, Quasi-variational inequalities: Semismooth Newton methods, optimal control, and uncertainties, Workshop on ``One World Optimization Seminar in Vienna'', June 3 - 7, 2024, Erwin Schrödinger International Institute for Mathematics and Physics and University of Vienna, Austria, June 4, 2024.

• M. Hintermüller, Risk-averse optimal control of random elliptic VIs, MS43 2024 SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS43: ``Efficient Solution Schemes for Optimization of Complex Systems Under Uncertainty'', February 27 - March 1, 2024, Trieste, Italy, February 27, 2024.

• I. Papadopoulos, A semismooth Newton method for obstacle--type quasivariational inequalities (online talk), MATH+ Spotlight talk (online event), November 6, 2024.

Preprints im Fremdverlag

• K. Knook, S. Olver, I. Papadopoulos, Quasi-optimal complexity hp-FEM for Poisson on a rectangle, Preprint no. arXiv.2402.11299, Cornell University, 2024, DOI 10.48550/arXiv.2402.11299 .

• P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, G. Zöttl, A Cournot-Nash model for a coupled hydrogen and electricity market, Preprint no. arxiv:2410.20534, Cornell University, 2024, DOI 10.48550/arXiv.2410.20534 .
  Abstract
  We present a novel model of a coupled hydrogen and electricity market on the intraday time scale, where hydrogen gas is used as a storage device for the electric grid. Electricity is produced by renewable energy sources or by extracting hydro- gen from a pipeline that is shared by non-cooperative agents. The resulting model is a generalized Nash equilibrium problem. Under certain mild assumptions, we prove that an equilibrium exists. Perspectives for future work are presented.

• I. Papadopoulos, S. Olver, A sparse hierarchical hp-finite element method on disks and annuli, Preprint no. arXiv.2402.12831, Cornell University, 2024, DOI 10.48550/arXiv.2402.12831 .