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.

• I. Papadopoulos, T. Gutleb, J. Carrillo, S. Olver, A frame approach for equations involving the fractional Laplacian, IMA Journal of Numerical Analysis, 00, pp. 1--29 (published online on 08.11.2025), DOI 10.1093/imanum/draf086 .
  Abstract
  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.

• I. Papadopoulos, S. Olver, A sparse hierarchical hp--finite element method on disks and annuli, Journal of Scientific Computing, 104 (2025), pp. 51/1--51/38, DOI 10.1007/s10915-025-02964-4 .
  Abstract
  We develop a sparse hierarchical hp-finite element method (hp-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse hp-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. We consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, the time-dependent Schrödinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.

• L. Banz, M. Hintermüller, A. Schröder, hp--finite elements for elliptic optimal control problems with control constraints, Computers & Mathematics with Applications. An International Journal, 196 (2025), pp. 288--311, DOI 10.1016/j.camwa.2025.07.030 .
  Abstract
  A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. The adjoint control is associated with the residual of the variational inequality but does not appear in the weak formulation. Each of the three variables is discretized independently by hp-finite elements. In particular, the non-penetration condition of the control variable is relaxed to a finite set of quadrature points. Sufficient conditions for the unique existence of a discrete solution are stated. Also a priori error estimates and guaranteed convergence rates are derived in terms of the mesh size as well as of the polynomial degree. Moreover, reliable and efficient a posteriori error estimates are presented, which enable hp-adaptive mesh refinements. Several numerical experiments demonstrate the applicability of the discretization with hp-finite elements, the efficiency of the a posteriori error estimates and the improvements with respect to the convergence order resulting from the application of hp-adaptivity. In particular, the hp-adaptive schemes lead to superior convergence properties.

• R.C. Contreras, M. Viana, E.S. Fonseca, M. Bongarti, Ö. Toygar, R.C. Guido, Exploring multicepstral features in a new classical machine learning--based framework for replay attack detection, Computers and Electrical Engineering, 127, Part B, pp. 110570/1--110570/xx (available online 27.07.2025), DOI 10.1016/j.compeleceng.2025.110570 .
  Abstract
  The integration of Internet of Things (IoT) technologies has accelerated the adoption of recognition and authentication systems, offering seamless access across devices from smart homes to workplace systems. Among biometric traits, voice stands out due to its simplicity, cleanliness, low capture cost, uniqueness, and the extensive computational resources supporting it in the scientific literature. Recently, however, spoofing risks have emerged as a serious challenge to the security of voice-based systems. To counteract these threats without additional hardware, techniques analyzing inherent voice signal features have been developed. This paper introduces a new soft computing framework based on classical machine learning classifiers such as Support Vector Machine (SVM), Random Forest (RF), and Logistic Regression (LR), comprising Gaussian-noise-based data augmentation, extraction and fusion of multiple cepstral and non-cepstral features, and dimensionality reduction through Singular Value Decomposition (SVD). In particular, we explore eight distinct cepstral extraction techniques, exemplified by popular approaches such as MFCC and CQCC, and sixteen additional non-cepstral metrics such as Zero Crossing Rate (ZCR) and Harmonic-to-Noise Ratio (HNR). Additionally, we generalize cepstral pattern representation by proposing cepstral multiprojection, a novel strategy designed to systematically reduce the dimensionality and redundancy of multicepstral matrices, thereby enhancing discriminative power and computational efficiency. Evaluated with the ASVSpoof 2017 v2.0 competition benchmark, our approach achieved competitive results, reaching 5.14% equal error rate (EER) on the Dev set and 10.58% on the Eval set, presenting an effective, interpretable, and computationally efficient alternative to state-of-the-art methods for replay attack detection in voice authentication systems. These findings provide a reproducible, reconfigurable, and modular soft computing framework that is interpretable, hardware-independent, and suitable for real-world deployment in voice spoofing detection systems.

• 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, Computer Methods in Applied Mechanics and Engineering, 445 (2025), pp. 118181/x--118181/xx, DOI 10.1016/j.cma.2025.118181 .
  Abstract
  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.

• 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, 46 (2025), pp. 7/505--7/539, 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.

• G. Dong, M. Hintermüller, C. Sirotenko, Dictionary learning based regularization in quantitative MRI: A nested alternating optimization framework, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 41 (2025), pp. 085007/1--085007/47, DOI 10.1088/1361-6420/adef74 .
  Abstract
  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.

• K. Knook, S. Olver, I. Papadopoulos, Quasi--optimal complexity hp--FEM for the Poisson equation on a rectangle, IMA Journal of Numerical Analysis, (2025), pp. 1--30 (published online on 17.11.2025), DOI 10.1093/imanum/draf102 .
  Abstract
  We show, in one dimension, that an hp-Finite Element Method (hp-FEM) discretisation can be solved in optimal complexity because the discretisation has a special sparsity structure that ensures that the reverse Cholesky factorisation---Cholesky starting from the bottom right instead of the top left---remains sparse. Moreover, computing and inverting the factorisation may parallelise across different elements. By incorporating this approach into an Alternating Direction Implicit (ADI) method à la Fortunato and Townsend (2020) we can solve, within a prescribed tolerance, an hp-FEM discretisation of the (screened) Poisson equation on a rectangle with quasi-optimal complexity: O(N^2 log N) operations where N is the maximal total degrees of freedom in each dimension. When combined with fast Legendre transforms we can also solve nonlinear time-evolution partial differential equations in a quasi-optimal complexity of O(N^2 log^2 N) operations, which we demonstrate on the (viscid) Burgers' equation. We also demonstrate how the solver can be used as an effective preconditioner for PDEs with variable coefficients, including coefficients that support a singularity.

• 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, Nature Communications, 16 (2025), pp. 11074/1--11074/18, DOI 10.1038/s41467-025-66846-x .
  Abstract
  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.

• L.M. Souza, R. Guido, R.C. Contreras, M. Viana, M. Bongarti, Improving voice spoofing detection through extensive analysis of multicepstral feature reduction, Sensors, 25 (2025), pp. 4821/1--4821/26, DOI 10.3390/s25154821 .
  Abstract
  Voice biometric systems play a critical role in numerous security applications, including electronic device authentication, banking transaction verification, and confidential communications. Despite their widespread utility, these systems are increasingly targeted by sophisticated spoofing attacks that leverage advanced artificial intelligence techniques to generate realistic synthetic speech. Addressing the vulnerabilities inherent to voice-based authentication systems has thus become both urgent and essential. This study proposes a novel experimental analysis that extensively explores various dimensionality reduction strategies in conjunction with supervised machine learning models to effectively identify spoofed voice signals. Our framework involves extracting multicepstral features followed by the application of diverse dimensionality reduction methods, such as Principal Component Analysis (PCA), Truncated Singular Value Decomposition (SVD), statistical feature selection (ANOVA F-value, Mutual Information), Recursive Feature Elimination (RFE), regularization-based LASSO selection, Random Forest feature importance, and Permutation Importance techniques. Empirical evaluation using the ASVSpoof 2017 v2.0 dataset measures the classification performance with the Equal Error Rate (EER) metric, achieving values of approximately 10%. Our comparative analysis demonstrates significant performance gains when dimensionality reduction methods are applied, underscoring their value in enhancing the security and effectiveness of voice biometric verification systems against emerging spoofing threats.

• A. Alphonse, C. Geiersbach, M. Hintermüller, Th.M. Surowiec, Risk--averse optimal control of random elliptic variational inequalities, ESAIM. Control, Optimisation and Calculus of Variations, 31 (2025), pp. 71/1--71/38, DOI 10.1051/cocv/2025045 .
  Abstract
  We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem.

• A. Alphonse, C. Christof, M. Hintermüller, I. Papadopoulos, A globalized inexact semismooth Newton method for nonsmooth fixed-point equations involving variational inequalities, Comptes Rendus Mathematique. Academie des Sciences. Paris, 93 (2026), pp. 1--55 (published online 10.11.2025), DOI 10.1007/s10589-025-00722-8 .
  Abstract
  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, G. Wachsmuth, Subdifferentials and penalty approximations of the obstacle problem, SIAM Journal on Optimization, 35 (2025), pp. 3/2017--3/2039, DOI 10.1137/24M172202X .
  Abstract
  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 penalized 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, cdot ). 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, C.N. Rautenberg, G. Wachsmuth, Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability, and optimal control, Interfaces and Free Boundaries. Mathematical Analysis, Computation and Applications, 27 (2025), pp. 4/521--4/573, DOI 10.4171/IFB/545 .
  Abstract
  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.

• 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, Composites Part A: Applied Science and Manufacturing, 202 (2026), pp. 109458/1-109458/15 (published online 26.11.2025), DOI 10.1016/j.compositesa.2025.109458 .
  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: fully resolved models (FRM), numerical upscaling method (NUM), scale-bridging method using data-driven machine learning methods (SBM) and a hybrid dual-scale solver incorporating PINNs. The FRM provides the highest fidelity model by fully resolving the micro- and mesoscale structural geometries, but requires high computational effort. NUM is a fully numerical dual-scale approach that considers uniform microscale permeability but neglects the microscale structural variability. The SBM accounts for the variability through a segment-wise assigned microscale permeability, which is determined using the data-driven ML method. This method shows no significant improvements over NUM with roughly the same computational efficiency and modeling runtimes of 45 min per simulation. The newly developed hybrid dual-scale solver incorporating PINNs shows the potential to overcome the problem of data scarcity of the data-driven surrogate approaches, as well as incorporating data from both scales via the hybrid loss function. The hybrid framework advances permeability modeling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing, while its full potential awaits realization as physics-informed machine learning approaches continue to mature.

Contributions to Collected Editions

• R.C. Contreras, A.F. Campanharo, M. Viana, M. Bongarti, R. Guido, Dimensionality reduction in multicepstral features for voice spoofing detection: Case studies with singular value decomposition, genetic algorithm, and auto--encoder, in: ICAISC 2024: International Conference on Artificial Intelligence and Soft Computing., 2025, pp. 227--244, DOI 10.1007/978-3-031-84356-3_19 .

Preprints, Reports, Technical Reports

• M. Hintermüller, M. Hinze, D. Korolev, Layerwise goal-oriented adaptivity for neural ODEs: An optimal control perspective, Preprint no. 3254, WIAS, Berlin, 2026, DOI 10.20347/WIAS.PREPRINT.3254 .
  Abstract, PDF (707 kByte)
  In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls acting as coefficients and a specific loss function. We implement our approach on the basis of a DG(0) Galerkin discretization of the neural ODE, leading to an explicit Euler time marching scheme. For the optimization we use steepest descent. Finally, we apply our method to the construction of neural networks for the classification of data sets, where we present results for a selection of well known examples from the literature.

• M. Bongarti, M. Hintermüller, Structure versus regularity of set-valued maps in convex generalized Nash equilibrium problems in Banach spaces, Preprint no. 3246, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3246 .
  Abstract, PDF (324 kByte)
  A generalized Nash equilibrium problem (GNEP) in Banach space consists of N > 1 optimal control problems with couplings in both the objective functions and, most importantly, in the feasible sets. We address the existence of equilibria for convex GNEPs in Banach space. We show that the standard assumption of lower semicontinuity of the set-valued constraint maps - foundational in the current literature on GNEPs - can be replaced by graph convexity or the so-called Knaster--Kuratowski --Mazurkiewicz (KKM) property. Lower semicontinuity is often essential for obtaining upper semicontinuity of best response maps, crucial for the existence theory based on Kakutani--Fan fixed-point arguments. However, in function spaces or PDE-constrained settings, verifying lower semicontinuity becomes much more challenging (even in convex cases), whereas graph convexity, for example, is often straightforward to check. Our results unify several existence theorems in the literature and clarify the structural role of constraint maps. We also extend Rosen's uniqueness condition to Banach spaces using a multiplier bias framework. Additionally, we present a geometric counterpart to our analytic framework using preference maps. This geometric is intended as a complement to, rather than a replacement for, the analytic theory developed in the main body of the paper.

• R. Baraldi, M. Hintermüller, Q. Wang, A multilevel proximal trust--region method for nonsmooth optimization with applications, Preprint no. 3235, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3235 .
  Abstract, PDF (11 MByte)
  Many large-scale optimization problems arising in science and engineering are naturally defined at multiple levels of discretization or model fidelity. Multilevel methods exploit this hierarchy to accelerate convergence by combining coarse- and fine-level information, a strategy that has proven highly effective in the numerical solution of partial differential equations and related optimization problems. It turns out that many applications in PDE-constrained optimization and data science require minimizing the sum of smooth and nonsmooth functions. For example, training neural networks may require minimizing a mean squared error plus an L1-regularization to induce sparsity in the weights. Correspondingly, we introduce a multilevel proximal trust-region method to minimize the sum of a non-convex, smooth and a convex, nonsmooth function. Exploiting ideas from the multilevel literature allows us to reduce the cost of the step computation, which is a major bottleneck in single level procedures. Our work unifies theory behind the proximal trust-region methods and multilevel recursive strategies. We prove global convergence of our method in finite dimensional space and provide an efficient non-smooth subproblem solver. We show the efficiency and robustness of our algorithm by means of numerical examples in PDE constrained optimization and machine-learning.

• A. Alphonse, P. Dvurechensky, I. Papadopoulos, C. Sirotenko, LeAP--SSN: A semismooth Newton method with global convergence rates, Preprint no. 3217, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3217 .
  Abstract, PDF (2618 kByte)
  We propose LeAP-SSN (Levenberg?Marquardt Adaptive Proximal SemismoothNewton method), a semismooth Newton-type method with a simple, parameter-free globalisation strategy that guarantees convergence from arbitrary starting points in nonconvex settings to stationary points, and under a Polyak?Łojasiewicz condition, to a global minimum, in Hilbert spaces. The method employs an adaptive Levenberg?Marquardt regularisation for the Newton steps, combined with backtracking, and does not require knowledge of problem-specific constants. We establish global nonasymptotic rates: O(1/k) for convex problems in terms of objective values, O(1/sqrtk) under nonconvexity in terms of subgradients, and linear convergence under a Polyak?Łojasiewicz condition. The algorithm achieves superlinear convergence under mild semismoothness and Dennis?Moré or partial smoothness conditions, even for non-isolated minimisers. By combining strong global guarantees with superlinear local rates in a fully parameter-agnostic framework, LeAP-SSN bridges the gap between globally convergent algorithms and the fast asymptotics of Newton's method. The practical efficiency of the method is illustrated on representative problems from imaging, contact mechanics, and machine learning.

• A. Sander, M. Fröhlich, M. Eigel, J. Eisert, M. Ali, M. Hintermüller, R.M. Milbradt, R. Wille, Ch.B. Mendl, Quantum circuit simulation with a local time--dependent variational principle, Preprint no. 3213, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3213 .
  Abstract, PDF (1416 kByte)
  Classical simulations of quantum circuits are vital for assessing potential quantum advantage and benchmarking devices, yet they require sophisticated methods to avoid the exponential growth of resources. Tensor network approaches, in particular matrix product states (MPS) combined with the time-evolving block decimation (TEBD) algorithm, currently dominate large-scale circuit simulations. These methods scale efficiently when entanglement is limited but suffer rapid bond dimension growth with increasing entanglement and handle long-range gates via costly SWAP insertions. Motivated by the success of the time-dependent variational principle (TDVP) in many-body physics, we reinterpret quantum circuits as series of discrete time evolutions, using gate generators to construct an MPS-based circuit simulation via a local TDVP formulation. This addresses TEBD's key limitations by (1) naturally accommodating long-range gates and (2) optimally representing states on the MPS manifold. By diffusing entanglement more globally, the method suppresses local bond growth and reduces memory and runtime costs. We benchmark the approach on five 49-qubit circuits: three Hamiltonian circuits (1D open and periodic Heisenberg, 2D 7 x 7 Ising) and two algorithmic ones (quantum approximate optimization, hardware-efficient ansatz). Across all cases, our method yields substantial resource reductions over standard tools, establishing a new state-of-the-art for circuit simulation and enabling advances across quantum computing, condensed matter, and beyond.

• Z. Amer, A. Avdzhieva, M. Bongarti, P. Dvurechensky, P. Farrell, U. Gotzes, F.M. Hante, A. Karsai, S. Kater, M. Landstorfer, M. Liero, D. Peschka, L. Plato, K. Spreckelsen, J. Taraz, B. Wagner, Modeling hydrogen embrittlement for pricing degradation in gas pipelines, Preprint no. 3201, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3201 .
  Abstract, PDF (12 MByte)
  This paper addresses aspects of the critical challenge of hydrogen embrittlement in the context of Germany's transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany's future energy infrastructure.

• 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.

• 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.

• 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.

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.

• Q. Wang, Robust multilevel training of artificial neural networks, MATH+ Day, Urania Berlin, November 17, 2025.

• M. Fröhlich, Quantum circuit simulation with a local time--dependent variational principle, Quantum Techniques in Machine Learning (QTML), November 16 - 21, 2025, National University of Singapore, Singapore, November 18, 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, Hierarchical proximal Galerkin: a fast hp--FEM solver for variational problems with pointwise inequality constraints, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2025), Minisymposium MS 55-2 ``Advancements and Applications of Solvers for PDE Systems with Nonsmooth Structures'', September 1 - 5, 2025, Universität Heidelberg, September 4, 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.

• I. Papadopoulos, The latent variable proximal point algorithm for variational problems with inequality constraints, 30th Biennial Conference on Numerical Analysis, Minisymposium M22 ``Advancements and applications of solvers for PDE systems with nonsmooth structures", June 24 - 27, 2025, University of Strathclyde, Glasgow, UK, June 26, 2025.

• F. Sauer, A monotonicity--based semismooth Newton method for almost sure state constraints, European Conference on Computational Optimization (EUCCO 2025), Session D1 ``Optimization under Uncertainty", September 29 - October 1, 2025, Universität Klagenfurt, Austria, October 1, 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.

• F. Sauer, Equilibria for distributed multi--modal energy systems under uncertainty, MATH+ Day, Urania Berlin, November 17, 2025.

• F. Sauer, On solving generalized Nash equilibrium problems constrained by PDEs on networks under uncertainty, Mathematics for Smart Energy, November 4 - 6, 2025, WIAS Berlin, November 5, 2025.

• M. Brokate, Derivatives of rate-independent evolutions, European Conference on Computational Optimization (EUCCO 2025), Session Q2 ``Non-smooth Optimization", September 29 - October 1, 2025, Universität Klagenfurt, Austria, September 30, 2025.

• A. Alphonse, A neural network approach to learning solutions of a class of elliptic variational inequalities, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2025), Minisymposium MS 25-1 ``Numerical Methods for Moving Interface and Free Boundary Problems'', September 1 - 5, 2025, Universität Heidelberg, September 3, 2025.

• M. Hintermüller, A PINN-based multi-complexity solver as a PDE-constrained optimization problem, The Third HKSIAM Biennial Conference, Department of Mathematics, The Chinese University of Hong Kong, China, July 8, 2025.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, 30th Biennial Conference on Numerical Analysis, Minisymposium M22 ``Advancements and applications of solvers for PDE systems with nonsmooth structures", June 24 - 27, 2025, University of Strathclyde, Glasgow, UK, June 26, 2025.

• M. Hintermüller, A globalized inexact semismooth Newton method for a class of nonsmooth fixed point equations, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2025), Minisymposium MS 55-1 ``Advancements and Applications of Solvers for PDE Systems with Nonsmooth Structures'', September 1 - 5, 2025, Universität Heidelberg, September 3, 2025.

• M. Hintermüller, A neural network approach to learning solutions of a class of elliptic variational inequalities, ALOP International Workshop on Algorithmic Optimization 2025, September 23 - 26, 2025, Universität Trier, September 24, 2025.

• M. Hintermüller, A neural network approach to learning solutions of a class of elliptic variational inequalities, The Mathematics of Scientific Machine Learning and Digital Twins, November 20 - 24, 2025, International School of Mathematics "G. Stampacchia", Erice, Italy, November 21, 2025.

• M. Hintermüller, Quantitative magnetic resonance imaging: data-driven, physics integrated models, 12th Applied Inverse Problems Conference, July 28 - August 1, 2025, FGV EMAp - Escola de Matemática Aplicada, Rio de Janeiro, Brazil, July 29, 2025.

• D. Korolev, A hybrid physics--informed neural network based multiscale solver as a partial differential equation constrained optimization problem, ICCOPT 2025 -- 8th International Conference on Continuous Optimization, Session ``Learning in PDE--based optimization and control'', July 19 - 24, 2025, University of Southern California, Los Angeles, USA, July 23, 2025.

• D. Korolev, A 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, A 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.

• D. Korolev, Hybrid solver project, 2nd Applied Mathematics Symposium, Bosch Research Campus, Renningen, October 9, 2025.

• C. Sirotenko, A neural network approach to learning solutions of a class of elliptic variational inequalities, 30th Biennial Conference on Numerical Analysis, Minisymposium M22 ``Advancements and applications of solvers for PDE systems with nonsmooth structures", June 24 - 27, 2025, University of Strathclyde, Glasgow, UK, June 26, 2025.

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.