Publikationen

Artikel in Referierten Journalen

  • G. Hu, A. Rathsfeld, Radiation conditions for the Helmholtz equation in a half plane filled by inhomogeneous periodic material, Journal of Differential Equations, 388 (2024), pp. 215--252, DOI 10.1016/j.jde.2024.01.008 .
    Abstract
    In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method.

  • W. VAN Ackooij, R. Henrion, H. Zidani, Pontryagin's principle for some probabilistic control problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 90 (2024), pp. 5/1--5/36, DOI 10.1007/s00245-024-10151-4 .
    Abstract
    In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin's optimality principle.

  • TH. Eiter, R. Lasarzik, Existence of energy-variational solutions to hyperbolic conservation laws, Calculus of Variations and Partial Differential Equations, 63 (2024), pp. 103/1--103/40, DOI 10.1007/s00526-024-02713-9 .
    Abstract
    We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.

  • M. Drieschner, R. Gruhlke, Y. Petryna, M. Eigel, D. Hömberg, Local surrogate responses in the Schwarz alternating method for elastic problems on random voided domains, Computer Methods in Applied Mechanics and Engineering, 405 (2023), pp. 115858/1--115858/18, DOI 10.1016/j.cma.2022.115858 .
    Abstract
    Imperfections and inaccuracies in real technical products often influence the mechanical behavior and the overall structural reliability. The prediction of real stress states and possibly resulting failure mechanisms is essential and a real challenge, e.g. in the design process. In this contribution, imperfections in elastic materials such as air voids in adhesive bonds between fiber-reinforced composites are investigated. They are modeled as arbitrarily shaped and positioned. The focus is on local displacement values as well as on associated stress concentrations caused by the imperfections. For this purpose, the resulting complex random one-scale finite element model is numerically solved by a new developed surrogate model using an overlapping domain decomposition scheme based on Schwarz alternating method. Here, the actual response of local subproblems associated with isolated material imperfections is determined by a single appropriate surrogate model, that allows for an accelerated propagation of randomness. The efficiency of the method is demonstrated for imperfections with elliptical and ellipsoidal shape in 2D and 3D and extended to arbitrarily shaped voids. For the latter one, a local surrogate model based on artificial neural networks (ANN) is constructed. Finally, a comparison to experimental results validates the numerical predictions for a real engineering problem.

  • M. Ebeling-Rump, D. Hömberg, R. Lasarzik, On a two-scale phasefield model for topology optimization, Discrete and Continuous Dynamical Systems -- Series S, published online on 26.11.2023, DOI 10.3934/dcdss.2023206 .
    Abstract
    In this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg--Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system.

  • M. Gugat, H. Heitsch, R. Henrion, A turnpike property for optimal control problems with dynamic probabilistic constraints, Journal of Convex Analysis, 30 (2023), pp. 1025--1052.
    Abstract
    In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition, additive random perturbations in each step and a terminal condition for the expected values. We study optimal control problems where the objective function consists of a term of tracking type for the expected values and a control cost. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.

  • C. Heiss, I. Gühring, M. Eigel, Multilevel CNNs for parametric PDEs, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 24 (2023), pp. 373/1--373/42.
    Abstract
    We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision with the number of weights only depending logarithmically on the resolution of the finest mesh. As a consequence, approximation bounds for the solution of parametric PDEs by neural networks that are independent on the (stochastic) parameter dimension can be derived.

    The performance of the proposed method is illustrated on high-dimensional parametric linear elliptic PDEs that are common benchmark problems in uncertainty quantification. We find substantial improvements over state-of-the-art deep learning-based solvers. As particularly challenging examples, random conductivity with high-dimensional non-affine Gaussian fields in 100 parameter dimensions and a random cookie problem are examined. Due to the multilevel structure of our method, the amount of training samples can be reduced on finer levels, hence significantly lowering the generation time for training data and the training time of our method.

  • CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing high-dimensional Bermudan options with hierarchical tensor formats, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 14 (2023), pp. 383--406, DOI 10.1137/21M1402170 .

  • M. Eigel, N. Farchmin, S. Heidenreich, P. Trunschke, Adaptive nonintrusive reconstruction of solutions to high-dimensional parametric PDEs, SIAM Journal on Scientific Computing, 45 (2023), pp. A457--A479, DOI 10.1137/21M1461988 .
    Abstract
    Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.

  • M. Eigel, N. Farchmin, S. Heidenreich, P. Trunschke, Efficient approximation of high-dimensional exponentials by tensor networks, International Journal for Uncertainty Quantification, 13 (2023), pp. 25--51, DOI 10.1615/Int.J.UncertaintyQuantification.2022039164 .
    Abstract
    In this work a general approach to compute a compressed representation of the exponential exp(h) of a high-dimensional function h is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.

  • R. Henrion, A. Jourani, B.S. Mordukhovich, Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions, Journal of Differential Equations, 366 (2023), pp. 408--443, DOI https://doi.org/10.1016/j.jde.2023.04.010 .
    Abstract
    This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.

  • D. Hömberg, R. Lasarzik, L. Plato, On the existence of generalized solutions to a spatio-temporal predator-prey system with prey-taxis, Journal of Evolution Equations, 23 (2023), pp. 20/1--20/44, DOI 10.1007/s00028-023-00871-5 .
    Abstract
    In this paper we consider a pair of coupled non-linear partial differential equations describing the interaction of a predator-prey pair. We introduce a concept of generalized solutions and show the existence of such solutions in all space dimension with the aid of a regularizing term, that is motivated by overcrowding phenomena. Additionally, we prove the weak-strong uniqueness of these generalized solutions and the existence of strong solutions at least locally-in-time for space dimension two and three.

  • R. Lasarzik, M.E.V. Reiter, Analysis and numerical approximation of energy-variational solutions to the Ericksen--Leslie equations, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 184 (2023), pp. 11/1--11/44, DOI 10.1007/s10440-023-00563-9 .
    Abstract
    We define the concept of energy-variational solutions for the Ericksen--Leslie equations in three spatial dimensions. This solution concept is finer than dissipative solutions and satisfies the weak-strong uniqueness property. For a certain choice of the regularity weight, the existence of energy-variational solutions implies the existence of measure-valued solutions and for a different choice, we construct an energy-variational solution with the help of an implementable, structure-inheriting space-time discretization. Computational studies are performed in order to provide some evidence of the applicability of the proposed algorithm.

  • R. Lasarzik, On the existence of energy-variational solutions in the context of multidimensional incompressible fluid dynamics, Mathematical Methods in the Applied Sciences, 47 (2024), pp. 4319--4344 (published online on 20.12.2023), DOI 10.1002/mma.9816 .
    Abstract
    We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations.

Preprints, Reports, Technical Reports

  • M. Bachmayr, M. Eigel, H. Eisenmann, I. Voulis, A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields, Preprint no. 3112, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3112 .
    Abstract, PDF (3246 kByte)
    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.

  • D. Hömberg, R. Lasarzik, L. Plato, Existence of weak solutions to an anisotropic electrokinetic flow model, Preprint no. 3104, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3104 .
    Abstract, PDF (1762 kByte)
    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.

  • A. Rathsfeld, Convergence of the method of rigorous coupled-wave analysis for the diffraction by two-dimensional periodic surface structures, Preprint no. 3081, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3081 .
    Abstract, PDF (424 kByte)
    The scattering matrix algorithm is a popular numerical method to simulate the diffraction of optical waves by periodic surfaces. The computational domain is divided into horizontal slices and, by a domain decomposition method coupling neighbour slices over the common interface via scattering data, a clever recursion is set up to compute an approximate operator, mapping incoming waves into outgoing. Combining this scattering matrix algorithm with numerical schemes inside the slices, methods like rigorous coupled wave analysis and Fourier modal methods were designed. The key for the analysis is the scattering problem over the slices. These are scattering problems with a radiation condition generalized for inhomogeneous cover and substrate materials and were first analyzed in [7]. In contrast to [7], where the scattering matrix algorithm for transverse electric polarization was treated without full discretization (no approximation by truncated Fourier series), we discuss the more challenging case of transverse magnetic polarization and look at the convergence of the fully-discretized scheme, i.e., at the rigorous coupled wave analysis for a fixed slicing into layers with vertically invariant optical index.

  • M.J. Arenas, D. Hömberg, R. Lasarzik, Optimal beam forming for laser materials processing, Preprint no. 3080, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3080 .
    Abstract, PDF (2441 kByte)
    We investigate an optimal control problem related to laser material treatments such as welding, remelting, hardening, or the 3D printing of metal components. The mathematical model leads to the investigation of a quasilinear elliptic state system with additional non-monotone lower-oder terms. We analyze the state system, derive first order optimality conditions and show first results for beam shaping.

  • M. Eigel, Ch. Miranda, Functional SDE approximation inspired by a deep operator network architecture, Preprint no. 3079, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3079 .
    Abstract, PDF (755 kByte)
    We present a novel approach to solve Stochastic Differential Equations (SDEs) with Deep Neural Networks by a Deep Operator Network (DeepONet) architecture. The notion of Deep-ONets relies on operator learning in terms of a reduced basis. We make use of a polynomial chaos expansion (PCE) of stochastic processes and call the corresponding architecture SDEONet. The PCE has been used extensively in the area of uncertainty quantification with parametric partial differential equations. This however is not the case with SDE, where classical sampling methods dominate and functional approaches are seen rarely. A main challenge with truncated PCEs occurs due to the drastic growth of the number of components with respect to the maximum polynomial degree and the number of basis elements. The proposed SDEONet architecture aims to alleviate the issue of exponential complexity by learning a sparse truncation of the Wiener chaos expansion. A complete convergence analysis is presented, making use of recent Neural Network approximation results. Numerical experiments illustrate the promising performance of the suggested approach in 1D and higher dimensions.

  • D. Sommer, R. Gruhlke, M. Kirstein, M. Eigel, C. Schillings, Generative modelling with tensor train approximations of Hamilton--Jacobi--Bellman equations, Preprint no. 3078, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3078 .
    Abstract, PDF (2117 kByte)
    Sampling from probability densities is a common challenge in fields such as Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular sampling tool. In [5] the authors point out that these log-densities can be obtained by solution of a Hamilton-Jacobi-Bellman (HJB) equation known from stochastic optimal control. While this HJB equation is usually treated with indirect methods such as policy iteration and unsuper-vised training of black-box architectures like Neural Networks, we propose instead to solve the HJB equation by direct time integration, using compressed polynomials represented in the Tensor Train (TT) format for spatial discretization. Crucially, this method is sample-free, agnostic to normalization constants and can avoid the curse of dimensionality due to the TT compression. We provide a complete derivation of the HJB equation?s action on Tensor Train polynomials and demonstrate the performance of the proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions.

  • M. Eigel, Ch. Miranda, J. Schütte, D. Sommer, Approximating Langevin Monte Carlo with ResNet-like neural network architectures, Preprint no. 3077, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3077 .
    Abstract, PDF (795 kByte)
    We sample from a given target distribution by constructing a neural network which maps samples from a simple reference, e.g. the standard normal distribution, to samples from the target. To that end, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, we show approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks and derive expressivity results for approximating the sample to target distribution map.

  • N. Ouanes, T. González Grandón, H. Heitsch, R. Henrion, Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints, Preprint no. 3069, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3069 .
    Abstract, PDF (1681 kByte)
    In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classical power system's operation tasks. Three alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The results show the dispatch modifications for battery and Diesel reserve planning, with the JCC model providing the most robust results, albeit with a small increase in costs.

  • H. Heitsch, R. Henrion, C. Tischendorf, Probabilistic maximization of time-dependent capacities in a gas network, Preprint no. 3066, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3066 .
    Abstract, PDF (421 kByte)
    The determination of free technical capacities belongs to the core tasks of a gas network owner. Since gas loads are uncertain by nature, it makes sense to understand this as a probabilistic problem as far as stochastic modeling of available historical data is possible. Future clients, however, don?t have a history or they do not behave in a random way, as is the case, for instance, in gas reservoir management. Therefore, capacity maximization turns into an optimization problem with uncertainty-related constrained which are partially of probabilistic and partially of robust (worst case) type. While previous attempts to solve this problem had be devoted to models with static (time-independent) gas flow, we aim at considering here transient gas flow subordinate to a PDE (Euler equations). The basic challenge here is two-fold: first, a proper way of joining probabilistic constraints to the differential equations has to be found. This will be realized on the basis of the so-called spherical-radial decomposition of Gaussian random vectors. Second, a suitable characterization of the worst-case load behaviour of future customers has to be figured out. It will be shown, that this is possible for quasi-static flow and can be transferred to the transient case. The complexity of the problem forces us to constrain ourselves in this first analysis to simple pipes or to a V-like structure of the network. Numerical solutions are presented and show that the differences between quasi-static and transient solutions are small, at least in these elementary examples.

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

  • P. Trunschke, M. Eigel, A. Nouy, Weighted sparsity and sparse tensor networks for least squares approximation, Preprint no. 3049, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3049 .
    Abstract, PDF (954 kByte)
    The approximation of high-dimensional functions is a ubiquitous problem in many scientific fields that is only feasible practically if advantageous structural properties can be exploited. One prominent structure is sparsity relatively to some basis. For the analysis of these best n-term approximations a relevant tool is the Stechkin's lemma. In its standard form, however, this lemma does not allow to explain convergence rates for a wide range of relevant function classes. This work presents a new weighted version of Stechkin's lemma that improves the best n-term rates for weighted ℓp-spaces and associated function classes such as Sobolev or Besov spaces. For the class of holomorphic functions, which for example occur as solutions of common high-dimensional parameter dependent PDEs, we recover exponential rates that are not directly obtainable with Stechkin's lemma. This sparsity can be used to devise weighted sparse least squares approximation algorithms as known from compressed sensing. However, in high-dimensional settings, classical algorithms for sparse approximation suffer the curse of dimensionality. We demonstrate that sparse approximations can be encoded efficiently using tensor networks with sparse component tensors. This representation gives rise to a new alternating algorithm for best n-term approximation with a complexity scaling polynomially in n and the dimension. We also demonstrate that weighted ℓpsummability not only induces sparsity of the tensor but also low ranks. This is not exploited by the previous format. We thus propose a new low-rank tensor train format with a single weighted sparse core tensor and an ad-hoc algorithm for approximation in this format. To analyse the sample complexity for this new model class we derive a novel result of independent interest that allows to transfer the restricted isometry property from one set to another sufficiently close set. We then prove that the new model class is close enough to the set of weighted sparse vectors such that the restricted isometry property transfers. Numerical examples illustrate the theoretical results for a benchmark problem from uncertainty quantification. Although they lead up to the analysis of our final model class, our contributions on weighted Stechkin and the restricted isometry property are of independent interest and can be read independently.

  • A. Agosti, R. Lasarzik, E. Rocca, Energy-variational solutions for viscoelastic fluid models, Preprint no. 3048, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3048 .
    Abstract, PDF (329 kByte)
    In this article, we introduce the concept of energy-variational solutions for a large class of systems of nonlinear evolutionary partial differential equations. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme. Weak-strong uniqueness follows by a suitable relative energy inequality.
    Our main motivation is to apply the general framework to viscoelastic fluid models. Therefore, we give a short overview on different versions of such models and their derivation. The abstract result is applied to two of these viscoelastic fluid models in full detail. In the conclusion, we comment on further applications of the general theory and its possible impact.

  • M. Eigel, N. Hegemann, Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients, Preprint no. 3036, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3036 .
    Abstract, PDF (394 kByte)
    Solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical results by providing a quasi-error reduction results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement.

  • C. Geiersbach, R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Preprint no. 3021, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3021 .
    Abstract, PDF (355 kByte)
    In this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.

Vorträge, Poster

  • J. Schütte, M. Eigel, Adaptive multilevel neural networks for parametric PDEs with error estimation, ICLR Workshop on AI4DifferentialEquations In Science, Vienna, Austria, May 11, 2024.

  • J. Schütte, Approximating Langevin Monte Carlo with resnet-like neural network architectures, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 25.02 ``Various Topics in Computational and Mathematical Methods in Datascience'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 20, 2024.

  • J. Schütte, Bayes for parametric PDEs with normalizing flows, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS37 ``Forward and Inverse Uncertainty Quantification for Nonlinear Problems -- Part II'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, Italy, February 27, 2024.

  • V. Aksenov, Learning distributions with regularized JKO scheme and low-rank tensor decompositions, Workshop on Optimal Transport from Theory to Applications, Berlin, March 11 - 15, 2024.

  • V. Aksenov, Learning distributions with regularized JKO scheme and low-rank tensor decompositions, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS153 ``Low Rank Methods for Random Dynamical Systems and Sequential Data Assimilation -- Part I'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, Italy, February 29, 2024.

  • V. Aksenov, Modelling distributions with Wasserstein proximal methods and low-rank tensor decompositions, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 15.06 ``UQ -- Sampling and Rare Events Estimation'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 21, 2024.

  • CH. Miranda, Functional SDE approximation inspired by a deep operator network architecture, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 25.02 ``Various Topics in Computational and Mathematical Methods in Datascience'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 19, 2024.

  • CH. Miranda, Solving stochastic differential equations using deep operator networks, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS25 ``Nonlinear Approximation of High-Dimensional Functions: Compositional, Low-Rank and Sparse Structures -- Part II'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, Italy, February 27, 2024.

  • D. Sommer, Approximating Langevin Monte Carlo with ResNet-like neural network architecture (online talk), University of Tokyo, Graduate School of Mathematical Sciences, Japan, March 13, 2024.

  • D. Sommer, Approximating Langevin Monte Carlo with resnet-like neural network architectures, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS25 ``Nonlinear Approximation of High-Dimensional Functions: Compositional, Low-Rank and Sparse Structures -- Part II'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, Italy, February 27, 2024.

  • M. Eigel, Adaptive multilevel neural networks for parametric PDEs with error control, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 15.05 ``Methodologies for Forward UQ'', March 18 - 22, 2024, Otto-von-Guericke-Universität Magdeburg, March 20, 2024.

  • M. Eigel, An operator network architecture for functional SDE representations, SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS6 ``Operator Learning in Uncertainty Quantification -- Part I'', February 27 - March 1, 2024, Savoia Excelsior Palace Trieste and Stazione Marittima, Italy, February 27, 2024.

  • M. Eigel, Generative modelling with tensor compressed HJB approximations, 11th International Conference "Inverse Problems: Modeling and Simulation" (IPMS 2024), Minisymposium M7 ``Bayesian, Variational, and Optimization Techniques for Inverse Problems in Stochastic PDEs'', May 26 - June 1, 2024, Paradise-Bay Hotel, Cirkewwa, Malta, May 31, 2024.

  • R. Henrion, D. Hömberg, N. Kliche, Optimal operation of mini-grids including battery management under uncertainty, 5. Workshop ``Women in Optimization 2024'', Friedrich-Alexander-Universität Erlangen, April 10 - 12, 2024.

  • R. Henrion, Chance constraints in energy management and aspects of nonsmoothness, Workshop ``Variational Analysis and Applications for Modeling of Energy Exchange'' (VAME 2024), May 13 - 14, 2024, Universität Trier, May 13, 2024.

  • R. Henrion, On a chance-constrained optimal control problem with turnpike property, 3rd International Conference on Variational Analysis and Optimization, January 16 - 19, 2024, Universidad de Chile, Santiago, Chile, January 16, 2024.

  • R. Lasarzik, Minimizing movements for damped Hamiltonian systems, Workshop on Optimal Transport from Theory to Applications, Berlin, March 11 - 15, 2024.

  • A. Rathsfeld, Analysis of scattering matrix algorithm for diffraction by periodic structures (online talk), University of Tokyo, Graduate School of Mathematical Sciences, Japan, March 13, 2024.

  • N. Kliche, A numerical approach for the optimal operation of mini-grids under uncertainty, 22nd European Conference on Mathematics for Industry (ECMI2023), MS17 ``ECMI SIG: Mathematics for the Digital Factory'', June 26 - 30, 2023, Wrocław University of Science and Technology Congress Centre, Poland, June 26, 2023.

  • L. Plato, R. Lasarzik, D. Hömberg, E. Emmrich, Nonlinear electrokinetics in anisotropic microfluids -- Analysis, simulation, and optimal control, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • L. Plato, Existence of weak solutions to an anisotropic electrokinetic flow model, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00874 ``Recent Advances in the Analysis and Numerics for Phase-Field Models'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

  • J. Schütte, A neural network architecture for sampling, WIAS Days, March 6 - 7, 2024, Humboldt-Universität zu Berlin, March 6, 2024.

  • J. Schütte, Adaptive multilevel neural networks for parametric PDEs with error estimation, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00960 ``Hierarchical Low Rank Tensors and DNNs for High-dimensional Approximation'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 23, 2023.

  • J. Schütte, Adaptive neural networks for parametric PDEs, 5th International Conference on Uncertainty Quantification in Computational Science and Engineering (UNCECOMP 2023), MS9 ``UQ and Data Assimilation with Sparse, Low-rank Tensor, and Machine Learning Methods'', June 12 - 14, 2023, Athens, Greece, June 14, 2023.

  • J. Schütte, Adaptive neural networks for parametric PDEs, Mini-Workshop ``Nonlinear Approximation of High-dimensional Functions in Scientific Computing, October 15 - 20, 2023, Mathematisches Forschungsinstitut Oberwolfach, October 20, 2023.

  • J. Schütte, Neural and tensor networks for parametric PDEs and inverse problems, Annual Meeting of SPP 2298, November 5 - 8, 2023, Evangelische Akademie, Tutzing, November 8, 2023.

  • J. Schütte, Neural networks as surrogates for parameter-dependent ordinary differential equations, ESGI 175 -- The Berlin Study Group with Industry, September 18 - 22, 2023, WIAS Berlin, September 22, 2023.

  • V. Aksenov, Simulation of Wasserstein gradient flows with low-rank tensor methods for sampling, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00837 ``Particle Methods for Bayesian Inference'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

  • H. Heitsch, Probabilistic maximization of time-dependent capacities in a gas network, Frontiers of Stochastic Optimization and its Applications in Industry, WIAS, Berlin, May 11, 2023.

  • CH. Miranda, Approximating Langevin Monte Carlo with ResNet-like neural network architectures, Mini-Workshop ``Nonlinear Approximation of High-dimensional Functions in Scientific Computing'', October 15 - 20, 2023, Mathematisches Forschungsinstitut Oberwolfach, October 18, 2023.

  • J. Polzehl, Smoothing techniques for quantitative MR, colloquium, Marquette University, Department of Mathematical and Statistical Sciences, Milwaukee, USA, November 3, 2023.

  • D. Sommer, Computing log-densities of reverse-time diffusion processes through Hamilton--Jacobi--Bellman equations (online talk), University of Tokyo, Graduate School of Mathematical Sciences, Japan, November 30, 2023.

  • D. Sommer, Computing log-densities of time-reversed diffusion processes through Hamilton--Jacobi--Bellman equations, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00837 ``Particle Methods for Bayesian Inference'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

  • D. Sommer, Industrial testing via reinforcement learning with tensor trains, ESGI 175 -- The Berlin Study Group with Industry, September 18 - 22, 2023, WIAS Berlin, September 22, 2023.

  • D. Sommer, Less interaction with forward models in Langevin dynamics, Centrale Nantes, Laboratoire de Mathématiques Jean Leray, France, June 27, 2023.

  • D. Sommer, Robust model predictive control for digital twins using feedback laws, Leibniz MMS Days 2023, April 17 - 19, 2023, Leibniz-Institut für Agrartechnik und Bioökonomie (ATB), Potsdam, April 18, 2023.

  • M. Eigel, M. Heida, M. Landstorfer, A. Selahi, Recovery of battery ageing dynamics with multiple timescales, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • M. Eigel, Accelerated interacting particle systems with low-rank tensor compression for Bayesian inversion, 5th International Conference on Uncertainty Quantification in Computational Science and Engineering (UNCECOMP 2023), MS9 ``UQ and Data Assimilation with Sparse, Low-rank Tensor, and Machine Learning Methods'', June 12 - 14, 2023, Athens, Greece, June 14, 2023.

  • M. Eigel, Accelerated interacting particle transport for Bayesian inversion, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00966 ``Theoretical and Computational Advances in Measure Transport'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 21, 2023.

  • M. Eigel, Convergence of adaptive empirical stochastic Galerkin FEM, ECCOMAS Young Investigators Conference (YIC2023), Minisymposium CAM01 ``Uncertainty Quantification of Differential Equations with Random Parameters: Methods and Applications'', June 19 - 21, 2023, University of Porto, June 19, 2023.

  • M. Eigel, Convergence of an empirical Galerkin method for parametric PDEs, SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS100 ``Randomized Solvers in Large-Scale Scientific Computing'', February 26 - March 3, 2023, Amsterdam, Netherlands, February 28, 2023.

  • M. Eigel, Convergence of empirical Galerkin FEM for parametric PDEs with sparse TTs, Universität Basel, Departement Mathematik und Informatik, Switzerland, May 12, 2023.

  • M. Eigel, Functional SDE approximation inspired by a deep operator network architecture, Mini-Workshop ``Nonlinear Approximation of High-dimensional Functions in Scientific Computing'', October 15 - 20, 2023, Mathematisches Forschungsinstitut Oberwolfach, October 18, 2023.

  • TH. Eiter, R. Lasarzik, Analysis of energy-variational solutions for hyperbolic conservation laws, Presentation of project proposals in SPP 2410 ``Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness'', Bad Honnef, April 28, 2023.

  • R. Henrion, D. Hömberg, N. Kliche, Modeling and simulation of mini-grids under uncertainty, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Tokyo, Japan, August 20 - 25, 2023.

  • R. Henrion, A control problem with random state constraints in probabilistic and almost-sure form, PGMO DAYS 2023, Session 10B ``Stochastic Optimization'', November 28 - 29, 2023, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF Lab Paris-Saclay, Palaiseau, France, November 29, 2023.

  • R. Henrion, Chance constraints in optimal control, ALOP colloquium, Universität Trier, Graduiertenkolleg ALOP, April 24, 2023.

  • R. Henrion, Existence and stability in controlled polyhedral sweeping processes (online talk), International Workshop on Nonsmooth Optimization: Theory, Algorithms and Applications (NOTAA2023) (Online Event), June 7 - 8, 2023, University of Isfahan, Iran, June 8, 2023.

  • R. Henrion, Optimality conditions for a PDE-constrained control problem with probabilistic and almost-sure state constraints, Nonsmooth And Variational Analysis (NAVAL) Conference, June 26 - 28, 2023, Université de Bourgogne, Dijon, France, June 27, 2023.

  • R. Henrion, Probabilistic constraints in optimal control, SIAM Conference on Optimization (OP23), MS 163 ``Risk Models in Stochastic Optimization'', May 31 - June 3, 2023, Seattle, USA, June 1, 2023.

  • R. Henrion, Turnpike phenomenon in discrete-time optimal control with probabilistic constraint, 2nd Vienna Workshop on Computational Optimization, March 15 - 17, 2023, Universität Wien, Austria, March 15, 2023.

  • D. Hömberg, R. Henrion, N. Kliche, Modeling and simulation of weakly coupled mini-grids under uncertainty, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

  • D. Hömberg, ODE-constrained optimal control, preparatory winter school within the project ``Ethiopian Norwegian Network in Computational Mathematics'' (ENNCoMat), November 27 - December 1, 2023, Addis Ababa University, Ethiopia.

  • D. Hömberg, On two-scale topology optimization for AM, The Fourth International Conference on Simulation for Additive Manufacturing (Sim-AM 2023), IS14 ``Advanced Methods and Innovative Technologies for the Optimal Design of Structures and Materials II'', July 26 - 28, 2023, Galileo Science Congress Center Munich-Garching, Garching, July 27, 2023.

  • D. Hömberg, Phase-field based topology optimization, Norwegian Workshop on Mathematical Optimization, Nonlinear and Variational Analysis 2023, April 26 - 28, 2023, Norwegian University of Science and Technology, Trondheim, Norway, April 27, 2023.

  • D. Hömberg, Two-scale topology optimization -- A phase field approach, 22nd European Conference on Mathematics for Industry (ECMI2023), MS17 ``ECMI SIG: Mathematics for the Digital Factory'', June 26 - 30, 2023, Wrocław University of Science and Technology Congress Centre, Poland, June 26, 2023.

  • D. Hömberg, Two-scale topology optimization for 3D printing, SIAM Conference on Computational Science and Engineering (CSE23), Minisymposium MS328 ``ECMI: Perspectives and Successes of Mathematical Challenges in Industrial Applications'', February 26 - March 3, 2023, Amsterdam, Netherlands, March 2, 2023.

  • R. Lasarzik, Analysis of an Allen--Cahn system in two scale topology optimization, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00874 ``Recent Advances in the Analysis and Numerics for Phase-Field Models'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

  • R. Lasarzik, Energy-variational solutions for conservation laws, Università degli Studi di Pavia, Dipartimento di Matematica, Italy, February 7, 2023.

  • R. Lasarzik, Energy-variational solutions for conservation laws, CRC Colloquium, Freie Universität Berlin, Department of Mathematics and Computer Science, October 19, 2023.

  • R. Lasarzik, Energy-variational solutions for different viscoelastic fluid models, Workshop ``Energetic Methods for Multi-Component Reactive Mixtures Modelling, Stability, and Asymptotic Analysis'', September 13 - 15, 2023, WIAS Berlin, September 15, 2023.

  • R. Lasarzik, Solvability for viscoelastic materials via the energy-variational approach, DMV Annual Meeting 2023, September 25 - 28, 2023, Technische Universität Ilmenau, September 25, 2023.

  • A. Rathsfeld, Analysis of scattering matrix algorithm, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00297 ``Wave Scattering Problems: Numerical Methods with Applications'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

  • A. Rathsfeld, Analysis of scattering matrix algorithm for diffraction by periodic surface structures, Chinese Academy of Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Beijing, China, September 4, 2023.

  • A. Rathsfeld, Plane-wave scattering by biperiodic gratings and rough surfaces: Radiation condition and far-field model, Nankai University, Department of Scientific and Engineering Computing, Tianjin, China, September 7, 2023.

  • A. Rathsfeld, Plane-wave scattering by biperiodic gratings and rough surfaces: Radiation condition and far-field model (online talk), University of Tokyo, Graduate School of Mathematical Sciences, Japan, October 26, 2023.

  • A. Rathsfeld, Simulation and inverse problems for rough surfaces by numerical methods of periodic grating structures (online talk), University of Tokyo, Graduate School of Mathematical Sciences, Japan, November 29, 2023.

Preprints im Fremdverlag

  • M. Eigel, J. Schütte, Adaptive multilevel neural networks for parametric PDEs with error estimation, Preprint no. arXiv:2403.12650, , 2024, DOI 10.48550/arXiv.2403.12650 .

  • P. Trunschke, A. Nouy, M. Eigel, Weighted sparsity and sparse tensor networks for least squares approximation, Preprint no. arXiv:2310.08942, Cornell University, 2023, DOI 10.48550/arXiv.2310.08942 .