Nonsmooth Variational Problems and Operator Equations

Publications

Articles in Refereed Journals

• M. Bongarti, Ch. Parkinson, W. Wang, Optimal control of a reaction-diffusion epidemic model with non-compliance, European Journal of Applied Mathematics, pp. 1--26 (published online on 14.04.2025), DOI 10.1017/S0956792525000130 .
  Abstract
  In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first?order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

• G. Dong, M. Hintermüller, K. Papafitsoros, K. Völkner, First-order conditions for the optimal control of learning-informed nonsmooth PDEs, Numerical Functional Analysis and Optimization. An International Journal, (2025), pp. 1--35 (published online on 23.04.2025), DOI 10.1080/01630563.2025.2488796 .
  Abstract
  In 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.

• 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.

• C. Geiersbach, R. Henrion, P. Pérez-Aroz, Numerical solution of an optimal control problem with probabilistic and almost sure state constraints, Journal of Optimization Theory and Applications, 204 (2025), pp. 7/1--7/30 (published online on 27.12.2024), DOI 10.1007/s10957-024-02578-0 .
  Abstract
  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.

• G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, GAMM-Mitteilungen, 48 (2025), pp. e202470014/1-- e202470014/35 (published online on 06.11.2024), 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. 69/1--69/45, 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, 203 (2024), pp. 165--195, 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, 157 (2025), pp. 213--248 (published online on 19.11.2024), DOI 10.1007/s00211-024-01438-3 .

Contributions to Collected Editions

• M.S. Viana, R.C. Contreras, P.C. Pessoa, M. Bongarti, H. Zamani, R.C. Guido, O. Morandin Junior, Massive conscious neighborhood--based crow search algorithm for the pseudo-coloring problem, in: Advances in Swarm Intelligence. ICSI 2024, Y. Tan, Y. Shi, eds., 14788 of Lecture Notes in Computer Science, Springer, Singapore, 2024, pp. 182-196, DOI 10.1007/978-981-97-7181-3_15 .

• R.C. Contreras, G.L. Heck, M.S. Viana, M. Bongarti, H. Zamani, R.C. Guido, Metaheuristic algorithms for enhancing multicepstral representation in voice spoofing detection: An experimental approach, in: Advances in Swarm Intelligence. ICSI 2024, Y. Tan, Y. Shi, eds., 14788 of Lecture Notes in Computer Science, Springer, Singapore, 2024, pp. 247--262, DOI 10.1007/978-981-97-7181-3_20 .

Preprints, Reports, Technical Reports

• I. Papadopoulos, Hierarchical proximal Galerkin: A fast hp--FEM solver for variational problems with pointwise inequality constraints, Preprint no. 3189, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3189 .
  Abstract, PDF (47 MByte)
  We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024), a recently introduced mesh-independent algorithm, to obtain a high-order finite element solver for variational problems with pointwise inequality constraints. This is achieved by discretizing the saddle point systems, arising from the latent variable proximal point method, with the hierarchical p-finite element basis. This results in discretized sparse Newton systems that admit a simple and effective block preconditioner. The solver can handle both obstacle-type and gradient-type constraints. We apply the resulting algorithm to solve obstacle problems with hp-adaptivity, a gradient-type constrained problem, and the thermoforming problem, an example of an obstacle-type quasi-variational inequality. We observe hp-robustness in the number of Newton iterations and only mild growth in the number of inner Krylov iterations to solve the Newton systems. Crucially we also provide wall-clock timings that are faster than low-order discretization counterparts.

• J.S. Dokken, P.E. Farrell, B. Keith, I. Papadopoulos, Th.M. Surowiec, The latent variable proximal point algorithm for variational problems with inequality constraints, Preprint no. 3188, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3188 .
  Abstract, PDF (17 MByte)
  The latent variable proximal point (LVPP) algorithm is a framework for solving infinite-dimensional variational problems with pointwise inequality constraints. The algorithm is a saddle point reformulation of the Bregman proximal point algorithm. At the continuous level, the two formulations are equivalent, but the saddle point formulation is more amenable to discretization because it introduces a structure-preserving transformation between a latent function space and the feasible set. Working in this latent space is much more convenient for enforcing inequality constraints than the feasible set, as discretizations can employ general linear combinations of suitable basis functions, and nonlinear solvers can involve general additive updates. LVPP yields numerical methods with observed mesh-independence for obstacle problems, contact, fracture, plasticity, and others besides; in many cases, for the first time. The framework also extends to more complex constraints, providing means to enforce convexity in the Monge?Ampère equation and handling quasi-variational inequalities, where the underlying constraint depends implicitly on the unknown solution. In this paper, we describe the LVPP algorithm in a general form and apply it to twelve problems from across mathematics.

• A. Alphonse, A. Djurdjevac, E. Engström, E. Hansen, Transmission problems and domain decompositions for non--autonomous parabolic equations on evolving domains, Preprint no. 3187, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3187 .
  Abstract, PDF (400 kByte)
  Parabolic equations on evolving domains model a multitude of applications including various industrial processes such as the molding of heated materials. Such equations are numerically challenging as they require large-scale computations and the usage of parallel hardware. Domain decomposition is a common choice of numerical method for stationary domains, as it gives rise to parallel discretizations. In this study, we introduce a variational framework that extends the use of such methods to evolving domains. In particular, we prove that transmission problems on evolving domains are well posed and equivalent to the corresponding parabolic problems. This in turn implies that the standard non-overlapping domain decompositions, including the Robin-Robin method, become well defined approximations. Furthermore, we prove the convergence of the Robin?Robin method. The framework is based on a generalization of fractional Sobolev-Bochner spaces on evolving domains, time-dependent Steklov-Poincaré operators, and elements of the approximation theory for monotone maps.

• I. Papadopoulos, T. Gutleb, J. Carrillo, S. Olver, A frame approach for equations involving the fractional Laplacian, Preprint no. 3177, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3177 .
  Abstract, PDF (699 kByte)
  Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, s ∈ (0, 1), on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to R^d, d ∈ 1, 2. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a 6^th-order Runge-Kutta time discretization), a fractional heat equation with a time-dependent exponent s(t), and a two-dimensional problem, observing spectral convergence in the spatial dimension for sufficiently smooth data.

• A. Sander, M. Fröhlich, M. Eigel, J. Eisert, P. Gelss, M. Hintermüller, R.M. Milbradt, R. Wille, Ch.B. Mendl, Large-scale stochastic simulation of open quantum systems, Preprint no. 3175, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3175 .
  Abstract, PDF (1495 kByte)
  Understanding the precise interaction mechanisms between quantum systems and their environment is crucial for advancing stable quantum technologies, designing reliable experimental frameworks, and building accurate models of real-world phenomena. However, simulating open quantum systems, which feature complex non-unitary dynamics, poses significant computational challenges that require innovative methods to overcome. In this work, we introduce the tensor jump method (TJM), a scalable, embarrassingly parallel algorithm for stochastically simulating large-scale open quantum systems, specifically Markovian dynamics captured by Lindbladians. This method is built on three core principles where, in particular, we extend the Monte Carlo wave function (MCWF) method to matrix product states, use a dynamic time-dependent variational principle (TDVP) to significantly reduce errors during time evolution, and introduce what we call a sampling MPS to drastically reduce the dependence on the simulation's time step size. We demonstrate that this method scales more effectively than previous methods and ensures convergence to the Lindbladian solution independent of system size, which we show both rigorously and numerically. Finally, we provide evidence of its utility by simulating Lindbladian dynamics of XXX Heisenberg models up to a thousand spins using a consumer-grade CPU. This work represents a significant step forward in the simulation of large-scale open quantum systems, with the potential to enable discoveries across various domains of quantum physics, particularly those where the environment plays a fundamental role, and to both dequantize and facilitate the development of more stable quantum hardware.

• 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.

• 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.

Talks, Poster

• Q. Wang, A multilevel proximal trust-region method for nonsmooth optimization with applications to scientific machine learning, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 25.08 ``Machine Learning and Data Science in Applied Mathematics and Mechanics'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 11, 2025.

• I. Papadopoulos, A sparse hierarchical hp-finite element method on disks, annuli, and cylinders, SIAM Conference on Computational Science and Engineering (CSE25), Minisymposium MS195 ``Advances in Domain Decomposition Methods and Fast Solvers Part II of II", March 3 - 7, 2025, Fort Worth, USA, March 5, 2025.

• I. Papadopoulos, Hierarchical proximal Galerkin: a fast hp-FEM solver for variational problems with pointwise inequality constraints, Scientific Computing Seminars, Brown University, Division of Applied Mathematics, Providence, Rhode Island, USA, April 11, 2025.

• I. Papadopoulos, Numerical analysis of a topology optimization problem for the compliance of a linearly elastic structure, Brown University, Division of Applied Mathematics, Providence, Rhode Island, USA, April 17, 2025.

• F. Sauer, A novel distributed method for PDE-constrained GNEPs, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 19.03 ``Optimisation of differential equations'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 9, 2025.

• D. Korolev, Hybrid physics-informed neural network based multiscale solver for multi-fidelity upscaling operator learning, DTE & AICOMAS 2025, 3rd IACM Digital Twins in Engineering (DTE 2025) & 1st ECCOMAS Artificial Intelligence and Computational Methods in Applied Sciences (AICOMAS 2025), February 17 - 21, 2025, Arts et Métiers -- ENSAM (Paris Campus), Paris, France, February 19, 2025.

• D. Korolev, Hybrid physics-informed neural network based multiscale solver for numerical simulations of composite materials, Leibniz MMS Days 2025, March 26 - 28, 2025, The Leibniz Research Network "Mathematical Modeling and Simulation", Leibniz Institute for Baltic Sea Research Warnemünde (IOW), March 26, 2025.

• Q. Wang, Robust multilevel training of artificial neural networks, MATH+ Day, Urania 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 Network ``Mathematical Modeling and Simulation,'' Leibniz-Institut für Verbundwerkstoffe GmbH (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 (8th Firedrake user and developer workshop), September 16 - 18, 2024, University of Oxford, UK, September 18, 2024.

• F. Sauer, Equilibria for distributed multi-modal energy systems under uncertainty, MATH+ Day, Urania Berlin, October 18, 2024.

• M. Brokate, Derivatives of rate-independent evolutions, 31th IFIP TC7 Conference 2024 on 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, Università degli Studi 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, 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): 77th Course of the International School of Mathematics ``Guido Stampacchia'', September 1 - 7, 2024, 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'', 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, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS43 ``Efficient Solution Schemes for Optimization of Complex Systems Under Uncertainty'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, 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.

External Preprints

• D. Korolev, T. Schmidt, D. Natarajan, S. Cassola, D. May, M. Duhovic, M. Hintermüller, Hybrid machine learning based scale bridging framework for permeability prediction of fibrous structures, Preprint no. arXiv:2502.05044, Cornell University, 2025, DOI 10.48550/arXiv.2502.05044 .
  Abstract
  This study introduces a hybrid machine learning-based scale-bridging framework for predicting the permeability of fibrous textile structures. By addressing the computational challenges inherent to multiscale modeling, the proposed approach evaluates the efficiency and accuracy of different scale-bridging methodologies combining traditional surrogate models and even integrating physics-informed neural networks (PINNs) with numerical solvers, enabling accurate permeability predictions across micro- and mesoscales. Four methodologies were evaluated: Single Scale Method (SSM), Simple Upscaling Method (SUM), Scale-Bridging Method (SBM), and Fully Resolved Model (FRM). SSM, the simplest method, neglects microscale permeability and exhibited permeability values deviating by up to 150% of the FRM model, which was taken as ground truth at an equivalent lower fiber volume content. SUM improved predictions by considering uniform microscale permeability, yielding closer values under similar conditions, but still lacked structural variability. The SBM method, incorporating segment-based microscale permeability assignments, showed significant enhancements, achieving almost equivalent values while maintaining computational efficiency and modeling runtimes of 45 minutes per simulation. In contrast, FRM, which provides the highest fidelity by fully resolving microscale and mesoscale geometries, required up to 270 times more computational time than SSM, with model files exceeding 300 GB. Additionally, a hybrid dual-scale solver incorporating PINNs has been developed and shows the potential to overcome generalization errors and the problem of data scarcity of the data-driven surrogate approaches. The hybrid framework advances permeability modelling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing.

• 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 .