Nonsmooth Variational Problems and Operator Equations

Publications

Monographs

• M. Brokate, J. Zimmer, F. Lindemann, Analysis 1 -- Ein zuverlässiger und verständlicher Begleiter für Studium und Prüfung, Springer Spektrum, Berlin, Heidelberg, 2023, XII, 298 pages, (Monograph Published), DOI 10.1007/978-3-662-67776-6 .

Articles in Refereed Journals

• 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. 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.

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

• A. Alphonse, D. Caetano, A. Djurdjevac, Ch.M. Elliot, Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs, Journal of Differential Equations, 353 (2023), pp. 268-338, DOI 10.1016/j.jde.2022.12.032 .
  Abstract
  We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev?Bochner spaces. An Aubin?Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev?Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.

• M. Flaschel, H. Yu, N. Reiter, J. Hinrichsen, S. Budday, P. Steinmann, S. Kumar, L. De Lorenzis, Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID, Journal of the Mechanics and Physics of Solids, 180 (2023), pp. 105404/1--105404/23, DOI 10.1016/j.jmps.2023.105404 .

• M. Brokate, C. Christof, Strong stationarity conditions for optimal control problems governed by a rate-independent evolution variational inequality, SIAM Journal on Control and Optimization, 61 (2023), pp. 2222--2250, DOI 10.1137/22M1494403 .

• M. Gugat, J. Habermann, M. Hintermüller, O. Huber, Constrained exact boundary controllability of a semilinear model for pipeline gas flow, European Journal of Applied Mathematics, 34 (2023), pp. 532--553, DOI 10.1017/S0956792522000389 .
  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.

• A. Kofler, F. Altekrüger, F.A. Ba, Ch. Kolbitsch, E. Papoutsellis, D. Schote, C. Sirotenko, F.F. Zimmermann, K. Papafitsoros, Learning regularization parameter-maps for variational image reconstruction using deep neural networks and algorithm unrolling, SIAM Journal on Imaging Sciences, 16 (2023), pp. 2202--2246, DOI 10.1137/23M1552486 .

• E. Marino, M. Flaschel, S. Kumar, L. De Lorenzis, Automated identification of linear viscoelastic constitutive laws with EUCLID, Mechanics of Materials, 181 (2023), pp. 104643/1--104643/12, DOI 10.1016/j.mechmat.2023.104643 .

• C. Geiersbach, T. Scarinci, A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty, Journal of Differential Equations, 364 (2023), pp. 635-666, DOI 10.1016/j.jde.2023.04.034 .

• Q. Wang, D. Yang, Y. Zhang, Real-variable characterizations and their applications of matrix-weighted Triebel--Lizorkin spaces, Journal of Mathematical Analysis and Applications, 529 (2024), pp. 127629/1--127629/37 (published online on 26.07.2023), DOI 10.1016/j.jmaa.2023.127629 .

• M. Hintermüller, T. Keil, Strong stationarity conditions for the optimal control of a Cahn--Hilliard--Navier--Stokes system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 89 (2024), pp. 12/1--12/28 (published online on 05.12.2023), DOI 10.1007/s00245-023-10063-9 .
  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 variational inequality of fourth order and the Navier-Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.

• M. Hintermüller, A. Kröner, Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier--Stokes equations with mixed-boundary conditions in a channel, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 87 (2023), pp. 15/1--15/38, DOI 10.1007/s00245-022-09938-0 .
  Abstract
  In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 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 generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 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.

Contributions to Collected Editions

• R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: 1st Conference on Research Data Infrastructure (CoRDI) -- Connecting Communities, Y. Sure-Vetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 69/1--69/4, DOI 10.52825/cordi.v1i.397 .
  Abstract
  MaRDI is building a research data infrastructure for mathematics and beyond based on semantic technologies (metadata, ontologies, knowledge graphs) and data repositories. Focusing on the algorithms, models and workflows, the MaRDI infrastructure will connect with other disciplines and NFDI consortia on data processing methods, solving real world problems and support mathematicians on research datamanagement

Preprints, Reports, Technical Reports

• 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.
  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.

• C. Geiersbach, R. Henrion, P. Pérez-Aroz, Numerical solution of an optimal control problem with probabilistic and almost sure state constraints, Preprint no. 3062, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3062 .
  Abstract, PDF (779 kByte)
  We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau-Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated.

• M. Hintermüller, D. Korolev, A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem, Preprint no. 3052, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3052 .
  Abstract, PDF (1045 kByte)
  In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multiscale PDEs. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. We propose a function space framework for the problem and develop an algorithm for its numerical solution, combining an adjoint-based technique from optimal control with automatic differentiation. The multiscale solver is applied to a heat transfer problem with oscillating coefficients, where the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties and accuracy of the PINN method. The relevance of the hybrid solver to numerical homogenization is discussed.

• C. Geiersbach, T. Suchan, K. Welker, Optimization of piecewise smooth shapes under uncertainty using the example of Navier--Stokes flow, Preprint no. 3037, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3037 .
  Abstract, PDF (1911 kByte)
  We investigate a complex system involving multiple shapes to be optimized in a domain, taking into account geometric constraints on the shapes and uncertainty appearing in the physics. We connect the differential geometry of product shape manifolds with multi-shape calculus, which provides a novel framework for the handling of piecewise smooth shapes. This multi-shape calculus is applied to a shape optimization problem where shapes serve as obstacles in a system governed by steady state incompressible Navier--Stokes flow. Numerical experiments use our recently developed stochastic augmented Lagrangian method and we investigate the choice of algorithmic parameters using the example of this application.

• C. Geiersbach, R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Preprint no. 3021, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3021 .
  Abstract, PDF (355 kByte)
  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.

Talks, Poster

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

• 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 Institute Berlin (ZIB), June 7, 2024.

• 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.

• C. Geiersbach, Basics of random algorithms, part 2, TRR 154 summer school on ``Optimization, Uncertainty and AI'', August 7 - 9, 2024, Universität Hamburg, August 8, 2024.

• C. Geiersbach, Numerical Solution of An Optimal Control Problem with Probabilistic or Almost Sure State Constraints, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS63 ``Efficient Solution Schemes for Optimization of Complex Systems Under Uncertainty'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, 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, 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, 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, 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.

• S. Essadi, A deterministic nonsmooth mean field game with control and state constraints, 9th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO'23), April 26 - 28, 2023, American University of Sharjah, UAE, Marrakesh, Morocco, April 27, 2023.

• S. Essadi, On nonsmooth mean field games with control and state constraints, SIAM Conference on Optimization (OP23), MS90 ``On Addressing Nonsmoothness, Hierarchy, and Uncertainty in Optimization and Games'', May 31 - June 3, 2023, Seattle, USA, June 1, 2023.

• Q. Wang, Robust multilevel training of artificial neural networks, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

• M. Bongarti, Network boundary control of the semilinear isothermal Euler equation modeling gas transport on a network of pipelines, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session S19 ``Optimization of Differential Equations'', May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• D. Korolev, ML4SIM: Mathematical Architecture, ML4SIM Consortium Meeting, WIAS, Berlin, November 15, 2023.

• D. Korolev, Machine learning for simulation intelligence in composite process design, Leibniz MMS Days 2023, Potsdam, April 17 - 19, 2023.

• D. Korolev, Physics-informed neural control of partial differential equations with applications to numerical homogenization, Kaiserslautern Applied and Industrial Mathematics Days -- KLAIM 2023, September 25 - 27, 2023, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Kaiserslautern, September 26, 2023.

• M. Brokate, Derivatives and optimal control of a scalar sweeping process, Variational Analysis and Optimization Seminar, University of Michigan, Ann Arbor, USA, March 31, 2023.

• M. Brokate, Derivatives and optimal control of a sweeping process, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session S19 ``Optimization of Differential Equations'', May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• M. Brokate, Strong stationarity conditions for an optimal control problem involving a rate-independent variational inequality, International Conference on Optimization: SIGOPT 2023, March 14 - 16, 2023, Brandenburgische Technische Universität Cottbus-Senftenberg, March 15, 2023.

• C. Geiersbach, Optimality Conditions in Control Problems with Probabilistic State Constraints, International Conference Stochastic Programming 2023, July 24 - 28, 2023, University of California, Davis, USA, July 25, 2023.

• C. Geiersbach, Optimality conditions for problems with probabilistic state constraints, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, Berlin, April 25, 2023.

• C. Geiersbach, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Frontiers of Stochastic Optimization and its Applications in Industry, May 10 - 12, 2023, WIAS, Berlin, May 11, 2023.

• C. Geiersbach, Optimization with random state constraints in probabilistic or almost-sure form, Thematic Einstein Semester Mathematical Optimization for Machine Learning, Summer Semester 2023, September 13 - 15, 2023, Zuse Instutite Berlin (ZIB), Berlin, September 15, 2023.

• C. Geiersbach, Optimization with random uniform state constraints, Optimal Control Theory and Related Fields, December 4 - 7, 2023, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile, December 6, 2023.

• C. Geiersbach, Stochastic approximation for shape optimization under uncertainty, Seminar in Numerical Analysis, Universität Basel, Switzerland, December 15, 2023.

• A. Alphonse, Analysis of a quasi-variational contact problem arising in thermoelasticity, European Conference on Computational Optimization (EUCCO), Session ``Non-smooth Optimization'', September 25 - 27, 2023, Universität Heidelberg, September 25, 2023.

• P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, Equilibria for distributed multi-modal energy systems under uncertainty, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network Informed PDEs based on approximate directional derivatives, FoCM 2023 -- Foundations of Computational Mathematics, Session II.2: ``Continuous Optimization'', June 12 - 21, 2023, Sorbonne University, Paris, France, June 15, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS390 ``Algorithms for Applications in Nonconvex, Nonsmooth Optimization'', February 26 - March 3, 2023, Amsterdam, Netherlands, March 3, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Conference on Optimization (OP23), MS 35 ``PDE-Constrained Optimization with Nonsmooth Structures or under Uncertainty'', May 31 - June 3, 2023, Seattle, USA, May 31, 2023.

• M. Hintermüller, Learning-informed and PINN-based multi scale PDE models in optimization, Conference on Deep Learning for Computational Physics, July 4 - 6, 2023, UCL -- London's Global University, UK, July 6, 2023.

• M. Hintermüller, Optimal control of (quasi)variational inequalities: Stationarity, risk-aversion, and numerical solution, Workshop on Optimization, Equilibrium and Complementarity, August 16 - 19, 2023, The Hong Kong Polytechnic University, Department of Applied Mathematic, August 19, 2023.

• M. Hintermüller, Optimal control of multiphase fluids and droplets (online talk), Workshop ``Control Methods in Hyperbolic Partial Differential Equations'' (Hybrid Event), November 5 - 10, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 7, 2023.

• M. Hintermüller, Short Course: Mathematics of PDE Constrained Optimization, Recent Trends in Optimization and Control: Short Course and Workshop, September 18 - 22, 2023, University of Pretoria, Future Africa Campus, South Africa, September 19, 2023.

• M. Hintermüller, PDE-constrained optimization with non-smooth learning-informed structures, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00711 ``Recent Advances in Optimal Control and Optimization'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 21, 2023.

• C. Sirotenko, Dictionary learning for an inverse problem in quantitative MRI, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00687 ``Recent advances in deep learning--based inverse and imaging problems'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

• C. Sirotenko, Machine Learning for Quantitative MRI, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, January 26, 2023.

External Preprints

• K. Knook, S. Olver, I.P.A. 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.P.A. 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 .

• M. Flaschel, H. Yu, N. Reiter, J. Hinrichsen, S. Budday, P. Steinmann, S. Kumar, L. De Lorenzis, Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID, Preprint no. arXiv:2305.16362, Cornell University, 2023, DOI 10.48550/arXiv.2305.16362 .

• J. Boddapati, M. Flaschel, S. Kumar, L. De Lorenzis, Ch. Daralo, Single-test evaluation of directional elastic properties of anisotropic structured materials, Preprint no. arXiv:2304.09112, Cornell University, 2023, DOI 10.48550/arXiv.2304.09112 .

• T.S. Gutleb, I.P.A. Papadopoulos, Explicit fractional Laplacians and Riesz potentials of classical functions, Preprint no. arXiv:2311.10896, Cornell University, 2023, DOI 10.48550/arXiv.2311.10896 .

• A. Kofler, F. Altekrüger, F.A. Ba, Ch. Kolbitsch, E. Papoutsellis, D. Schote, C. Sirotenko, F.F. Zimmermann, K. Papafitsoros, Learning regularization parameter-maps for variational image reconstruction using deep neural networks and algorithm unrolling, Preprint no. arXiv:2301.05888, Cornell University, 2023, DOI 10.48550/arXiv.2301.05888 .

• A. Kofler, F. Altekrüger, F.A. Ba, Ch. Kolbitsch, E. Papoutsellis, D. Schote, C. Sirotenko, F.F. Zimmermann, K. Papafitsoros, Unrolled three-operator splitting for parameter-map learning in low dose X-ray CT reconstruction, Preprint no. arXiv:2304.08350, Cornell University, 2023, DOI 10.48550/arXiv.2304.08350 .

• I.P.A. Papadopoulos, T.S. Gutleb, J.A. Carrillo, S. Olver, A frame approach for equations involving the fractional Laplacian, Preprint no. arXiv:2311.12451, Cornell University, 2023, DOI 10.48550/arXiv.2311.12451 .