Nonlinear Optimization and Inverse Problems

Publications

Monographs

• D. Peschka, M. Thomas, A. Zafferi, Reference map approach to Eulerian thermomechanics using GENERIC, J. Fuhrmann, D. Hömberg, W.H. Müller, W. Weiss, eds., 238 of Advanced Structured Materials, Springer Cham, 2025, pp. 39--70, (Chapter Published), DOI 10.1007/978-3-031-93918-1_3 .
  Abstract
  An Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordinates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e., the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated.

• P. Colli, J. Sprekels, Hyperbolic relaxation of the chemical potential in the viscous Cahn--Hilliard equation, J. Fuhrmann, D. Hömberg, W.H. Müller, W. Weiss, eds., 238 of Advanced Structured Materials, Springer Cham, 2025, pp. 529--556, (Chapter Published), DOI 10.1007/978-3-031-93918-1_18 .
  Abstract
  An Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordinates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e., the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated.

• M. Eigel, Ch. Merdon, Chapter Eight --- A posteriori error control for stochastic Galerkin FEM with high-dimensional random parametric PDEs, F. Chouly, S.P.A. Bordas, R. Becker, P. Omnes, eds., 60 of Advances in Applied Mechanics, Elsevier, 2025, pp. 347--397, (Chapter Published), DOI 10.1016/bs.aams.2025.02.008 .

• A. Mielke, An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC, J. Fuhrmann, D. Hömberg, W.H. Müller, W. Weiss, eds., 238 of Advanced Structured Materials, Springer Cham, 2025, pp. 13--38, (Chapter Published), DOI 10.1007/978-3-031-93918-1_2 .
  Abstract
  An Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordinates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e., the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated.

• J. Fuhrmann, D. Hömberg, W.H. Müller, W. Weiss, eds., Advances in Continuum Physics: In Memoriam Wolfgang Dreyer, 238 of Advanced Structured Materials, Springer, 2025, pp. vii--834, (Monograph Published), DOI 10.1007/978-3-031-93918-1 .

Articles in Refereed Journals

• V. Aksenov, M. Eigel, An Eulerian approach to regularized JKO scheme with low-rank tensor decompositions for Bayesian inversion, Journal of Scientific Computing, 103 (2025), paper no. 3, DOI 10.1007/s10915-025-03027-4 .
  Abstract
  The possibility of using the Eulerian discretization for the problem of modelling high dimensional distributions and sampling, is studied. The problem is posed as a minimization problem over the space of probability measures with respect to the Wasserstein distance and solved with the entropy-regularized JKO scheme. Each proximal step can be formulated as a fixed-point equation and solved with accelerated methods, such as Anderson's. The usage of the low-rank Tensor Train format allows to overcome the curse of dimensionality, i.e. the exponential growth of degrees of freedom with dimension, inherent to Eulerian approaches. The resulting method requires only pointwise computations of the unnormalized posterior and is, in particular, gradient-free. Fixed Eulerian grid allows to employ a caching strategy, significally reducing the expensive evaluations of the posterior. When the Eulerian model of the target distribution is fitted, the passage back to the Lagrangian perspective can also be made, allowing to approximately sample from the distribution. We test our method both for synthetic target distributions and particular Bayesian inverse problems and report comparable or better performance than the baseline Metropolis-Hastings MCMC with the same amount of resources. Finally, the fitted model can be modified to facilitate the solution of certain associated problems, which we demonstrate by fitting an importance distribution for a particular quantity of interest. We release our code at https://github.com/viviaxenov/rJKOtt.

• H. Heitsch, R. Henrion, On the Lipschitz continuity of the spherical cap discrepancy around generic point sets, Uniform Distribution Theory, 20 (2025), pp. 35--63, DOI 10.20347/WIAS.PREPRINT.3192 .
  Abstract
  The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy.

• M. Bachmayr, M. Eigel, H. Eisenmann, I. Voulis, A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields, SIAM Journal on Numerical Analysis, 63 (2025), pp. 1776--1807, DOI 10.1137/24M1649253 .
  Abstract
  The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.

• 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.

• R. Henrion, G. Stadler, F. Wechsung, Optimal control under uncertainty with joint chance state constraints: Almost-everywhere bounds, variance reduction, and application to (bi-)linear elliptic PDEs, SIAM/ASA Journal on Uncertainty Quantification, 13 (2025), pp. 1028--1053, DOI 10.1137/24M171557X .
  Abstract
  We study optimal control of PDEs under uncertainty with the state variable subject to joint chance constraints. The controls are deterministic, but the states are probabilistic due to random variables in the governing equation. Joint chance constraints ensure that the random state variable meets pointwise bounds with high probability. For linear governing PDEs and elliptically distributed random parameters, we prove existence and uniqueness results for almost-everywhere state bounds. Using the spherical-radial decomposition (SRD) of the uncertain variable, we prove that when the probability is very large or small, the resulting Monte Carlo estimator for the chance constraint probability exhibits substantially reduced variance compared to the standard Monte Carlo estimator. We further illustrate how the SRD can be leveraged to efficiently compute derivatives of the probability function, and discuss different expansions of the uncertain variable in the governing equation. Numerical examples for linear and bilinear PDEs compare the performance of Monte Carlo and quasi-Monte Carlo sampling methods, examining probability estimation convergence as the number of samples increases. We also study how the accuracy of the probabilities depends on the truncation of the random variable expansion, and numerically illustrate the variance reduction of the SRD.

• R. Henrion, D. Hömberg, N. Kliche, Modeling and simulation of an isolated mini-grid including battery operation strategies under uncertainty using chance constraints, Energy Systems, published online on 19.09.2025, DOI 10.1007/s12667-025-00764-1 .
  Abstract
  This paper addresses the challenge of handling uncertainties in mini-grid operation, crucial for achieving universal access to reliable and sustainable energy, especially in regions lacking access to a national grid. Mini-grids, consisting of small-scale power generation systems and distribution infrastructure, offer a cost-effective solution. However, the intermittency and uncertainty of renewable energy sources poses challenges, mitigated by employing batteries for energy storage. Optimizing the lifespan of the battery energy storage system is critical, requiring a balance between degradation and operational expenses, with battery operation strategies playing a key role in achieving this balance. Accounting for uncertainties in renewable energy sources, demand, and ambient temperature is essential for reliable energy management strategies. By formulating a probabilistic optimal control problem for minimizing the daily operational costs of stand-alone mini-grids under uncertainty, and exploiting the concept of joint chance constraints, we address the uncertainties inherent in battery dynamics and the associated operational constraints.

• D. Hömberg, R. Lasarzik, L. Plato, Existence of suitable weak solutions to an anisotropic electrokinetic flow model, Journal of Differential Equations, 428 (2025), pp. 511--584, DOI 10.1016/j.jde.2025.02.018 .
  Abstract
  In this article we present a system of coupled non-linear PDEs modelling an anisotropic electrokinetic flow. We show the existence of suitable weak solutions in three spatial dimensions, that is weak solutions which fulfill an energy inequality, via a regularized system. The flow is modelled by a Navier--Stokes--Nernst--Planck--Poisson system and the anisotropy is introduced via space dependent diffusion matrices in the Nernst--Planck and Poisson equation.

• R. Lasarzik, E. Rocca, R. Rossi, Existence and weak-strong uniqueness for damage systems in viscoelasticity, Nonlinearity, 38 (2025), pp. 125016/1--125016/59, DOI 10.1088/1361-6544/ae1433 .
  Abstract
  In this paper we investigate the existence of solutions and their weak-strong uniqueness property for a PDE system modelling damage in viscoelastic materials. In fact, we address two solution concepts, emphweak and emphstrong solutions. For the former, we obtain a global-in-time existence result, but the highly nonlinear character of the system prevents us from proving their uniqueness. For the latter, we prove local-in-time existence. Then, we show that the strong solution, as long as it exists, is unique in the class of weak solutions. This emphweak-strong uniqueness statement is proved by means of a suitable relative energy inequality.

Contributions to Collected Editions

• P. Hagemann, J. Schütte, D. Sommer, M. Eigel, G. Steidl, Sampling from Boltzmann densities with physics informed low-rank formats, Scale Space and Variational Methods in Computer Vision, 10th International Conference, SSVM 2025, Proceedings, Part I, Dartington, UK, May 18 - 22, 2025, 15667 of Lecture Notes in Computer Science (LNCS), Springer, 2025, pp. 374--386, DOI 10.1007/978-3-031-92366-1_29 .
  Abstract
  Our method proposes the efficient generation of samples from an unnormalized Boltzmann density by solving the underlying continuity equation in the low-rank tensor train (TT) format. It is based on the annealing path commonly used in MCMC literature, which is given by the linear interpolation in the space of energies. Inspired by Sequential Monte Carlo, we alternate between deterministic time steps from the TT representation of the flow field and stochastic steps, which include Langevin and resampling steps. These adjust the relative weights of the different modes of the target distribution and anneal to the correct path distribution. We showcase the efficiency of our method on multiple numerical examples.

Preprints, Reports, Technical Reports

• A. Goessmann, J. Schütte, M. Fröhlich, M. Eigel, A tensor network formalism for neuro-symbolic AI, Preprint no. 3257, WIAS, Berlin, 2026, DOI 10.20347/WIAS.PREPRINT.3257 .
  Abstract, PDF (689 kByte)
  The unification of neural and symbolic approaches to artificial intelligence remains a central open challenge. In this work, we introduce a tensor network formalism, which captures sparsity principles originating in the different paradigms in tensor decompositions. In particular, we describe a basis encoding scheme for functions and model neural decompositions as tensor decompositions. Furthermore, the proposed formalism can be applied to represent logical formulas and probability distributions as structured tensor decompositions. This unified treatment identifies tensor network contractions as a fundamental inference class and formulates efficiently scaling reasoning algorithms, originating from probability theory and propositional logic, as contraction message passing schemes. The framework enables the definition and training of hybrid logical and probabilistic models, which we call Hybrid Logic Networks. The theoretical concepts are accompanied by the python library tnreason, which enables the implementation and practical use of the proposed architectures.

• M. Eigel, Ch. Miranda, A. Nouy, D. Sommer, Approximation and learning with compositional tensor trains, Preprint no. 3253, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3253 .
  Abstract, PDF (1203 kByte)
  We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train format. This format can encode standard approximation tools, such as (sparse) polynomials, deep neural networks (DNNs) with fixed width, or tensor networks with arbitrary permutation of the inputs, or more general affine coordinate transformations, with similar complexities. This format can be viewed as a DNN with width exponential in the input dimension and structured weights matrices. Compared to DNNs, this format enables controlled compression at the layer level using efficient tensor algebra. par On the optimization side, we derive a layerwise algorithm inspired by natural gradient descent, allowing to exploit efficient low-rank tensor algebra. This relies on low-rank estimations of Gram matrices, and tensor structured random sketching. Viewing the format as a discrete dynamical system, we also derive an optimization algorithm inspired by numerical methods in optimal control. Numerical experiments on regression tasks demonstrate the expressivity of the new format and the relevance of the proposed optimization algorithms. par Overall, CTTs combine the expressivity of compositional models with the algorithmic efficiency of tensor algebra, offering a scalable alternative to standard deep neural networks.

• F. Cheng, R. Lasarzik, M. Thomas, Analysis of a Cahn--Hilliard model for viscoelastoplastic two-phase flows, Preprint no. 3247, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3247 .
  Abstract, PDF (431 kByte)
  We study a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids, which arises in geodynamics. A phase-field variable indicates the proportional distribution of the two fluids in the mixture. The motion of the incompressible mixture is described in terms of the volume-averaged velocity. Besides a volume-averaged Stokes-like viscous contribution, the Cauchy stress tensor in the momentum balance contains an additional volume-averaged internal stress tensor to model the elastoplastic behavior. This internal stress has its own evolution law featuring the nonlinear Zaremba-Jaumann time-derivative and the subdifferential of a non-smooth plastic potential. The well-posedness of this system is studied in two cases: Based on a regularization by stress-diffusion we obtain the existence of Leray-Hopf-type weak solutions. In order to deduce existence results also in the absence of the regularization, we introduce the concept of dissipative solutions, which is based on an estimate for the relative energy. We discuss general properties of dissipative solutions and show their existence for the viscoelastoplastic two-phase model in the setting of stress-diffusion. By a limit passage in the relative energy inequality for vanishing stress-diffusion, we conclude an existence result for the non-regularized model.

• D. Bernhard, H. Heitsch, R. Henrion, F. Liers, M. Stingl, A. Uihlein, V. Zipf, Continuous stochastic gradient and spherical radial decomposition, Preprint no. 3245, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3245 .
  Abstract, PDF (634 kByte)
  In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks.

• M. Eigel, L. Grasedyck, Th. Le, J. Schütte, Neural and tensor networks for high-dimensional parametric PDEs and sampling, Preprint no. 3223, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3223 .
  Abstract, PDF (16 MByte)
  In this chapter, we present various methods based on neural network architectures and tensor decompositions for addressing parametric partial differential equations (PDEs) and sampling from unnormalized densities in high-dimensional settings. Parametric PDEs arise in many areas of science and engineering, where one seeks to solve a PDE efficiently for a wide range of parameter values or where the data is inherently uncertain. Such problems have been analysed extensively in Uncertainty Quantification in recent years. Directly related and also highly relevant for practical applications is a class of statistical inverse problems. It can be approached in a Bayesian framework through sampling techniques. par Two main strategies are central to this chapter: deconstructing complex problems into manageable, simpler subproblems in a sensible way and designing problem-specific model architectures that suit the specific properties and requirements of the problem at hand. The developed methods are supported by theoretical convergence guarantees and demonstrate state-of-the-art performance in numerical experiments.

• 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.

• R. Lasarzik, L. Plato, A relative energy inequality for an anisotropic Navier--Stokes--Nernst--Planck--Poisson system --- Weak-strong uniqueness and a posteriori error estimates, Preprint no. 3208, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3208 .
  Abstract, PDF (1714 kByte)
  In this work, we build upon the framework of suitable weak solutions to the anisotropic Navier--Stokes--Nernst--Planck--Poisson (NSNPP) system, as developed in [HPL25], and establish a relative energy inequality for these solutions. This inequality serves as the basis for proving the weak-strong uniqueness property. Additionally, we exploit the relative energy inequality as a tool to obtain a posteriori error estimates. We are interested in the high-viscosity-low-Reynolds number limit of the NSNPP, which leads to the anisotropic Stokes--Nernst--Planck--Poisson (SNPP) system. Utilizing the relative energy framework, we derive an error estimate for the distance between solutions to the NSNPP and SNPP model in natural energy and dissipation norms. Moreover, we prove the existence of regular local-in-time solutions to the NSNPP as well as the SNPP system, possessing sufficient regularity to be admissible in the relative energy framework.

• 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.

• M. Eigel, C. Heiss, J. Schütte, Multi-level neural networks for high-dimensional parametric obstacle problems, Preprint no. 3193, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3193 .
  Abstract, PDF (766 kByte)
  A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface structure of the obstacle. As governing equation, a stationary elliptic diffusion problem is assumed. The high-dimensional solution of the obstacle problem is approximated by a specifically constructed convolutional neural network (CNN). This novel algorithm is inspired by a finite element constrained multigrid algorithm to represent the parameter to solution map. This has two benefits: First, it allows for efficient practical computations since multi-level data is used as an explicit output of the NN thanks to an appropriate data preprocessing. This improves the efficacy of the training process and subsequently leads to small errors in the natural energy norm. Second, the comparison of the CNN to a multigrid algorithm provides means to carry out a complete a priori convergence and complexity analysis of the proposed NN architecture. Numerical experiments illustrate a state-of-the-art performance for this challenging problem.

• R. Lasarzik, Energy-variational structure in evolution equations, Preprint no. 3185, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3185 .
  Abstract, PDF (325 kByte)
  We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered examples, we prove that a certain class of improved measure-valued solutions can be equivalently expressed as an energy-variational solution. The first concept represents the solution as a high-dimensional Young measure, whether for the second concept, only a scalar auxiliary variable is introduced and the formulation is relaxed to an energy-variational inequality. We investigate four examples: the two-phase Navier--Stokes equations, a quasilinear wave equation, a system stemming from polyconvex elasticity, and the Ericksen--Leslie equations equipped with the Oseen--Frank energy. The wide range of examples suggests that this is a recurrent feature in evolution equations in general.

• M. Eigel, Ch. Merdon, A posteriori error control for stochastic Galerkin FEM with high-dimensional random parametric PDEs, Preprint no. 3174, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3174 .
  Abstract, PDF (5084 kByte)
  PDEs with random data are investigated and simulated in the field of Uncertainty Quantification (UQ), where uncertainties or (planned) variations of coefficients, forces, domains and boundary con- ditions in differential equations formally depend on random events with respect to a pre-determined probability distribution. The discretization of these PDEs typically leads to high-dimensional (determin- istic) systems, where in addition to the physical space also the (often much larger) parameter space has to be considered. A proven technique for this task is the Stochastic Galerkin Finite Element Method (SGFEM), for which a review of the state of the art is provided. Moreover, important concepts and results are summarized. A special focus lies on the a posteriori error estimation and the derivation of an adaptive algorithm that controls all discretization parameters. In addition to an explicit residual based error estimator, also an equilibration estimator with guaranteed bounds is discussed. Under cer- tain mild assumptions it can be shown that the successive refinement produced by such an adaptive algorithm leads to a sequence of approximations with guaranteed convergence to the true solution. Nu- merical examples illustrate the practical behavior for some common benchmark problems. Additionally, an adaptive algorithm for a problem with a non-affine coefficient is shown. By transforming the original PDE a convection-diffusion problem is obtained, which can be treated similarly to the standard affine case.

Talks, Poster

• V. Aksenov, Accelerated fixed-point iteration on the Bures--Wasserstein manifold, Rheinisch-Westfälische Technische Hochschule Aachen, Institut für Geometrie und Praktische Mathematik, April 30, 2025.

• J. Schütte, Data handling for sedimentation studies for lake Abaya (online talk), Second Ethiopian Study Group with Industry, December 1 - 5, 2025, Addis Ababa Science and Technology University, Ethiopia, December 2, 2025.

• J. Schütte, Integration of logical tensor networks into LLMs for explainable and efficient reasoning, MATH+ Day, Urania Berlin, November 17, 2025.

• V. Aksenov, Learning distributions with regularized JKO scheme and low-rank tensor decompositions, Physikalisch-Technische Bundesanstalt, Berlin, July 9, 2025.

• A. Goessmann, Constructing quantum circuits for inference, Projekttreffen Q-Rom, Landsberg am Lech, October 23, 2025.

• A. Goessmann, Integration of logical tensor networks into LLMs for explainable and efficient reasoning, MATH+ Day, Urania Berlin, November 17, 2025.

• H. Heitsch, An enumerative formula for the spherical cap discrepancy, XVIIth Conference on Stochastic Programming (ICSP 2025), Invited Session ``Chance-Constrained Programming'', July 28 - August 1, 2025, Paris, France, July 30, 2025.

• H. Heitsch, Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints, Workshop ``Mathematics for Smart Energy'' (Ma4SmEn25), November 4 - 6, 2025, WIAS Berlin, November 6, 2025.

• M. Śliwiński, Analysis of energy-variational solutions for hyperbolic conservation laws, SPP 2410 Annual Statusseminar 2025, March 10 - 11, 2025, Darmstadt, March 10, 2025.

• M. Śliwiński, Analysis of energy-variational solutions to hyperbolic conservation laws with application to the Euler--Korteweg system, Workshop ``Mathematical Analysis of Fluid Flows by Variational Methods'' (MAFF 2025), September 29 - October 1, 2025, WIAS Berlin, September 29, 2025.

• L. Plato, A relative energy inequality for an anisotropic Navier--Stokes--Nernst--Planck--Poisson system -- Weak-strong uniqueness and a posteriori error estimates, 1st GAMM Activity Group Workshop ``Analysis of Partial Differential Equations and Calculus of Variations'', September 22 - 24, 2025, Technische Universität Dresden, September 22, 2025.

• L. Plato, A relative energy inequality for an anisotropic Navier--Stokes--Nernst--Planck--Poisson system --- Weak-strong uniqueness and a posteriori error estimates, Workshop ``Mathematical Analysis of Fluid Flows by Variational Methods'' (MAFF 2025), September 29 - October 1, 2025, WIAS Berlin, September 29, 2025.

• L. Plato, Biological pest control --- A spatio-temporal predator-prey model with prey-taxis, Norwegian Workshop on Mathematical Optimization, Nonlinear and Variational Analysis (MONVA), Norwegian University of Science and Technology, Norway, May 19 - 20, 2025.

• L. Plato, Existence and weak-strong uniqueness of suitable weak solutions to an anisotropic electrokinetic flow model, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.01 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 8, 2025.

• J. Schütte, Multilevel neural networks for high-dimensional parametric obstacle problems, The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) , Minisymposium ``MS55 Advancements and Applications of Solvers for PDE Systems with Nonsmooth Structures'', September 1 - 5, 2025, Universität Heidelberg, September 3, 2025.

• J. Schütte, Neural and tensor networks for parametric PDEs and inverse problems, Ludwig-Maximilians-Universität München, Mathematisches Institut, June 17, 2025.

• J. Schütte, Sampling from Boltzmann densities with physics informed low-rank formats, 10th International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2025), Dartington Hall, Totnes, UK, May 18 - 22, 2025.

• J. Breuer, A numerical analysis of energy-variational solutions for electroosmotic flow in nematic liquid crystals, October 13, 2025.

• M. Eigel, High-dimensional distribution sampling with applications in Bayesian inversion, Colloquium on Applied Mathematics, Universität Göttingen, Institut für Numerische Mathematik, February 4, 2025.

• M. Eigel, High-dimensional distribution sampling with applications in Bayesian inversion, Nankai University, School of Mathematical Sciences, Tianjin, China, June 11, 2025.

• M. Eigel, Implicit Wasserstein steps and Hamilton-Jacobi flows: Low-rank tensor sampling in high dimensions, Workshop ``Mathematics for Uncertainty Quantification", August 14 - 15, 2025, Aachen, August 14, 2025.

• M. Eigel, Low-rank tensor methods for high-dimensional measure transport and quantum circuit simulation, Beijing Academy of Sciences and Technology, China, June 16, 2025.

• M. Eigel, Properties and optimisation of compositional tensor networks, The 15th International Conference on Spectral and High Order Methods (ICOSAHOM 2025), Minisymposium MS 122 ``Tensor networks and compositional functions for high-dimensional approximation'', July 13 - 18, 2025, McGill University, Montréal, Canada, July 14, 2025.

• R. Henrion, Chance constraints in optimal control, Lothar-Collatz-Kolloquium für Angewandte Mathematik, Universität Hamburg, Fachbereich Mathematik, April 17, 2025.

• R. Henrion, Optimal control of polyhedral sweeping processes with chance constraints, XVIIth Conference on Stochastic Programming (ICSP 2025), Invited Session ``Chance-Constrained Programming'', July 28 - August 1, 2025, Paris, France, July 30, 2025.

• R. Henrion, Optimal control under uncertainty with joint chance state constraints: Almost-everywhere bounds, variance reduction, and application to (bi)linear elliptic PDEs, PGMO DAYS 2025, November 18 - 19, 2025, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF Lab Paris-Saclay, Palaiseau, France, November 19, 2025.

• R. Lasarzik, Energy-variational framework in fluid dynamics, Workshop ``Hydrodynamic Models and Multi-Scale Analysis in PDEs'', September 14 - 17, 2025, Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland, September 17, 2025.

• R. Lasarzik, Energy-variational structure in evolution, GRK 2339 IntComSin Colloquium Erlangen -- Regensburg, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, June 13, 2025.

• R. Lasarzik, Energy-variational structure in evolution equations, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2025), Section 14.02 ``Applied Analysis'', April 7 - 11, 2025, Poznan University of Technology, Poland, April 9, 2025.

• R. Lasarzik, The energy-variational structure in evolution, Thematic Programme on Free Boundary Problems, Erwin Schrödinger International Institute for Mathematics and Physics, Wien, Austria, September 30, 2025.

• R. Lasarzik, The minimizing movements scheme for hyperbolic conservation laws, The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) , Minisymposium MS14 ``Advances on Structure-Preserving Numerical Methods for Nonlinear PDEs'', September 1 - 5, 2025, Universität Heidelberg, September 1, 2025.

• R. Lasarzik, Weak solutions and weak-strong uniqueness for a Cahn--Hilliard--type model with chemotaxis, Workshop ``Free Boundaries and Growth'', December 9 - 12, 2025, Erwin Schrödinger International Institute for Mathematics and Physics, Wien, Austria, December 11, 2025.

• R. Lasarzik, tba, 24th GAMM Seminar on Microstructures 2025, January 30 - 31, 2025.