Nichtlineare Optimierung und Inverse Probleme

Publikationen

Artikel in Referierten Journalen

• 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. 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.

• 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.Uncertainty.Quantification.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, 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.

• H. Heitsch, R. Henrion, Th. Kleinert, M. Schmidt, On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints, Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, 84 (2022), pp. 651--685, DOI 10.1007/s10898-022-01161-z .
  Abstract
  Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.

• M. Branda, R. Henrion, M. Pištěk, Value at risk approach to producer's best response in electricity market with uncertain demand, Optimization. A Journal of Mathematical Programming and Operations Research, published online on 15.05.2022, DOI 10.1080/02331934.2022.2076232 .
  Abstract
  We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided.

• K. El Karfi, R. Henrion, D. Mentagui, An agricultural investment problem subject to probabilistic constraints, Computational Management Science, 19 (2022), pp. 683--701, DOI 10.1007/s10287-022-00431-1 .

• G. Thiele, Th. Johanni, D. Sommer, J. Krüger, Decomposition of a cooling plant for energy efficiency optimization using OptTopo, Energies, 15 (2022), pp. 8387/1--8387/16, DOI 10.3390/en15228387 .

• X. Yu, G. Hu, W. Lu, A. Rathsfeld, PML and high-accuracy boundary integral equation solver for wave scattering by a locally defected periodic surface, SIAM Journal on Numerical Analysis, 60 (2022), pp. 2592--2625, DOI 10.1137/21M1439705 .
  Abstract
  This paper studies the perfectly-matched-layer (PML) method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges to the true solution in the physical subregion of the strip with an error bounded by the reciprocal PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem over the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot converge exponentially to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip.

• M. Ebeling-Rump, D. Hömberg, R. Lasarzik, Two-scale topology optimization with heterogeneous mesostructures based on a local volume constraint, Computers & Mathematics with Applications. An International Journal, 126 (2022), pp. 100--114, DOI 10.1016/j.camwa.2022.09.004 .
  Abstract
  A new approach to produce optimal porous mesostructures and at the same time optimizing the macro structure subject to a compliance cost functional is presented. It is based on a phase-field formulation of topology optimization and uses a local volume constraint (LVC). The main novelty is that the radius of the LVC may depend both on space and a local stress measure. This allows for creating optimal topologies with heterogeneous mesostructures enforcing any desired spatial grading and accommodating stress concentrations by stress dependent pore size. The resulting optimal control problem is analysed mathematically, numerical results show its versatility in creating optimal macroscopic designs with tailored mesostructures.

• M. Eigel, R. Gruhlke, D. Moser, Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains, Computational Mechanics, 71 (2023), pp. 615--636 (published online on 27.12.2022), DOI 10.1007/s00466-022-02261-z .
  Abstract
  We develop a new fuzzy arithmetic framework for efficient possibilistic uncertainty quantification. The considered application is an edge detection task with the goal to identify interfaces of blurred images. In our case, these represent realisations of composite materials with possibly very many inclusions. The proposed algorithm can be seen as computational homogenisation and results in a parameter dependent representation of composite structures. For this, many samples for a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying effective material tensor map, an appropriate diffeomorphism is constructed to generate a family of meshes reflecting the possible material realisations. In the application, the uncertainty model is propagated through distance maps with respect to consecutive symmetry class tensors. Additionally, the efficacy of the best/worst estimate analysis of the homogenisation map as a bound to the average displacement for chessboard like matrix composites with arbitrary star-shaped inclusions is demonstrated.

• M. Eigel, R. Schneider, D. Sommer, Dynamical low-rank approximations of solutions to the Hamilton--Jacobi--Bellman equation, Numerical Linear Algebra with Applications, 30 (2023), pp. e2463/1--e2463/20 (published online on 03.08.2022), DOI 10.1002/nla.2463 .
  Abstract
  We present a novel method to approximate optimal feedback laws for nonlinar optimal control basedon low-rank tensor train (TT) decompositions. The approach is based on the Dirac-Frenkel variationalprinciple with the modification that the optimisation uses an empirical risk. Compared to currentstate-of-the-art TT methods, our approach exhibits a greatly reduced computational burden whileachieving comparable results. A rigorous description of the numerical scheme and demonstrations ofits performance are provided.

• M. Eigel, R. Gruhlke, M. Marschall, Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion, Statistics and Computing, 32 (2022), pp. 27/1--27/27, DOI 10.1007/s11222-022-10087-1 .
  Abstract
  A novel method for the accurate functional approximation of possibly highly concentrated probability densities is developed. It is based on the combination of several modern techniques such as transport maps and nonintrusive reconstructions of low-rank tensor representations. The central idea is to carry out computations for statistical quantities of interest such as moments with a convenient reference measure which is approximated by an numerical transport, leading to a perturbed prior. Subsequently, a coordinate transformation leads to a beneficial setting for the further function approximation. An efficient layer based transport construction is realized by using the Variational Monte Carlo (VMC) method. The convergence analysis covers all terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback-Leibler divergence. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a central motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity.

• M. Eigel, O. Ernst, B. Sprungk, L. Tamellini, On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion, SIAM Journal on Numerical Analysis, 60 (2022), pp. 659--687, DOI 10.1137/20M1364722 .
  Abstract
  Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting.

• M. Eigel, M. Haase, J. Neumann, Topology optimisation under uncertainties with neural networks, Algorithms, 15 (2022), pp. 241/1--241/34, DOI https://doi.org/10.3390/a15070241 .

• M. Eigel, R. Schneider, P. Trunschke, Convergence bounds for empirical nonlinear least-squares, ESAIM: Mathematical Modelling and Numerical Analysis, 56 (2022), pp. 79--104, DOI 10.1051/m2an/2021070 .
  Abstract
  We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.

• TH. Eiter, K. Hopf, R. Lasarzik, Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models, Advances in Nonlinear Analysis, 12 (2023), pp. 20220274/1--20220274/31 (published online on 03.10.2022), DOI 10.1515/anona-2022-0274 .
  Abstract
  We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and an internal stress. This stress tensor is transported via the Zaremba--Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray--Hopf type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the non-diffusive limit in the relative energy inequality satisfied by generalized solutions for non-zero stress diffusion.

• R. Lasarzik, E. Rocca, G. Schimperna, Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model, Rendiconti Lincei -- Matematica e Applicazioni, 33 (2022), pp. 229--269, DOI 10.4171/RLM/970 .
  Abstract
  In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.

Beiträge zu Sammelwerken

• M. Kirstein, D. Sommer, M. Eigel, Tensor-train kernel learning for Gaussian processes, in: Proceedings of the Eleventh Symposium on Conformal and Probabilistic Prediction with Applications, U. Johansson, H. Boström, K.A. Nguyen, Z. Luo, L. Carlsson, eds., 179 of Proceedings of Machine Learning Research, 2022, pp. 253--272.

• G. Thiele, Th. Johanni, D. Sommer, M. Eigel, J. Krüger, OptTopo: Automated set-point optimization for coupled systems using topology information, in: 2022 8th International Conference on Control, Decision and Information Technologies (CoDIT), IEEE, 2022, pp. 224--229, DOI 10.1109/CoDIT55151.2022.9803985 .

Preprints, Reports, Technical Reports

• 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. Heiss, I. Gühring, M. Eigel, Multilevel CNNs for parametric PDEs, Preprint no. 3035, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3035 .
  Abstract, PDF (1483 kByte)
  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.

• M. Ebeling-Rump, D. Hömberg, R. Lasarzik, On a two-scale phasefield model for topology optimization, Preprint no. 3026, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3026 .
  Abstract, PDF (2052 kByte)
  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.

• 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.

• A. Rathsfeld, Simulating rough surfaces by periodic and biperiodic gratings, Preprint no. 2989, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2989 .
  Abstract, PDF (1180 kByte)
  The scattering of acoustic and electro-magnetic plane waves by rough surfaces is the subject of many books and papers. For simplicity, we consider the special case, described by a Dirichlet boundary value problem of the Helmholtz equation in the half space above the surface. We recall the formulae of the far-field pattern and the far-field intensity. The far-field can be defined formally for general rough surfaces. However, the derivation as asymptotic limits works only for waves, which decay for surface points tending to infinity. Comparing with the case of periodic surface structures, it is clear that the rigorous model of plane-wave scattering is accurate for the near field close to the surface. For the far field, however, the finite extent of the beams in the planes orthogonal to the propagation direction is to be taken into account. Doing this rigorously, leads to extremely expensive computations or is simply impossible. Therefore and to enable the approximation of waves above the rough surface by waves above periodic and biperiodic rough structures, we consider a simplified model of beams. The beam is restricted to a cylindrical domain around a ray in propagation direction, and the wave is equal to a plane wave inside of this domain and to zero outside. Based on this beam model, we derive the corresponding asymptotic formulae for the wave and its intensity. The intensity is equal to the formally defined far-field intensity multiplied by a simple cosine factor. Under special assumptions, the intensity for the rough surface can be approximated by that for rough periodic and biperiodic surface structures. In particular, we can cope with the case of shallow roughness, where the reflected intensity includes, besides the smooth density function w.r.t. the angular direction, a plane-wave beam propagating into the reflection direction of the planar mirror.
  Altogether, the main point of the paper is to fix the technical assumptions needed for the far-field formula of a simple beam model and for the approximation by the far fields of periodized rough surfaces. Furthermore, using the beam model, we discuss numerical experiments for rough surfaces defined as realizations of a random field and, to get a more practical case, the Dirichlet condition is replaced by a transmission condition. The far-field intensity function for a rough surface is the limit of intensity functions for periodized rough surfaces if the period tends to infinity. However, almost the same intensity function can be obtained with a fixed period by computing the average over many different realizations of the random field. Finally, we present numerical results for an inverse problem, where the parameters of the random field are sought from measured mean values of the intensities.

• M. Eigel, R. Gruhlke, D. Sommer, Less interaction with forward models in Langevin dynamics, Preprint no. 2987, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2987 .
  Abstract, PDF (1423 kByte)
  Ensemble methods have become ubiquitous for the solution of Bayesian inference problems. State-of-the-art Langevin samplers such as the Ensemble Kalman Sampler (EKS), Affine Invariant Langevin Dynamics (ALDI) or its extension using weighted covariance estimates rely on successive evaluations of the forward model or its gradient. A main drawback of these methods hence is their vast number of required forward calls as well as their possible lack of convergence in the case of more involved posterior measures such as multimodal distributions. The goal of this paper is to address these challenges to some extend. First, several possible adaptive ensemble enrichment strategies that successively enlarge the number of particles in the underlying Langevin dynamics are discusses that in turn lead to a significant reduction of the total number of forward calls. Second, analytical consistency guarantees of the ensemble enrichment method are provided for linear forward models. Third, to address more involved target distributions, the method is extended by applying adapted Langevin dynamics based on a homotopy formalism for which convergence is proved. Finally, numerical investigations of several benchmark problems illustrates the possible gain of the proposed method, comparing it to state-of-the-art Langevin samplers.

• TH. Eiter, R. Lasarzik, Existence of energy-variational solutions to hyperbolic conservation laws, Preprint no. 2974, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2974 .
  Abstract, PDF (546 kByte)
  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.

• R. Gruhlke, M. Eigel, Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport, Preprint no. 2927, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2927 .
  Abstract, PDF (10 MByte)
  A unsupervised learning approach for the computation of an explicit functional representation of a random vector Y is presented, which only relies on a finite set of samples with unknown distribution. Motivated by recent advances with computational optimal transport for estimating Wasserstein distances, we develop a new Wasserstein multi-element polynomial chaos expansion (WPCE). It relies on the minimization of a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence.

  As a requirement for an efficient polynomial basis expansion, a suitable (minimal) stochastic coordinate system X has to be determined with the aim to identify ideally independent random variables. This approach generalizes representations through diffeomorphic transport maps to the case of non-continuous and non-injective model classes M with different input and output dimension, yielding the relation Y=M(X) in distribution. Moreover, since the used PCE grows exponentially in the number of random coordinates of X, we introduce an appropriate low-rank format given as stacks of tensor trains, which alleviates the curse of dimensionality, leading to only linear dependence on the input dimension. By the choice of the model class M and the smooth loss function, higher order optimization schemes become possible. It is shown that the relaxation to a discontinuous model class is necessary to explain multimodal distributions. Moreover, the proposed framework is applied to a numerical upscaling task, considering a computationally challenging microscopic random non-periodic composite material. This leads to tractable effective macroscopic random field in adopted stochastic coordinates.

Vorträge, Poster

• 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.

• 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.

• 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.

• 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, 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, Convergence of an empirical Galerkin method for parametric PDEs, SIAM Conference on Computational Science and Engineering (CSE23), MS100: ``Randomized Solvers in Large-Scale Scientific Computing (Part I)'', 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.

• 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, 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), MS163: ``Risk Models in Stochastic Optimization'', May 31 - June 3, 2023, The Sheraton Grand 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, 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), MS328: ECMI: ``Perspectives and Successes of Mathematical Challenges in Industrial Applications (Part II)'', February 26 - March 3, 2023, Amsterdam, Netherlands, March 2, 2023.

• L. Plato, Biological pest control -- Analysis and numerics for a spatio-temporal predator-prey system (online talk), Technische Universität Berlin, Institut für Mathematik, January 10, 2022.

• L. Plato, Generalized solutions in the context of a nonlocal predator-prey model (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Generalized Solvability Concepts for Evolutionary PDEs and their Properties'', March 14 - 18, 2022, March 16, 2022.

• J. Schütte, Adaptive neural networks for parametric PDE, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), PP04: ``Theoretical Foundations of Deep Learning'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• J. Schütte, Adaptive neural networks for parametric PDEs, Annual Meeting of SPP 2298, November 20 - 23, 2022, Evangelische Akademie, Tutzing, November 21, 2022.

• J. Schütte, Adaptive neural tensor networks for parametric PDEs, Workshop on the Approximation of Solutions of High-Dimensional PDEs with Deep Neural Networks within the DFG Priority Programme 2298 ``Theoretical Foundations of Deep Learning'', May 30 - 31, 2022, Universität Bayreuth, May 31, 2022.

• R. Gruhlke, Annual report 2022 -- MuScaBlaDes (subproject 4 within SPP1886), Jahrestreffen des SPP 1886, Rheinisch-Westfälische Technische Hochschule Aachen, August 17, 2022.

• R. Gruhlke, Wasserstein polynomial chaos expansion with application to computational homogenization and Baysian inference, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 15: ``Uncertainty Quantification'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• H. Heitsch, An algorithmic approach for solving optimization problems with probabilistic/robust (probust) constraints (online talk), TRR154 Summer School on Modelling, Simulation and Optimization for Energy Networks (Online Event), June 8 - 9, 2022, June 8, 2022.

• D. Sommer, Dynamical low rank approximation in molecular dynamics and optimal control, MASCOT-NUM 2022, June 7 - 9, 2022, Clermont Ferrand, France, June 7, 2022.

• D. Sommer, Less interaction with forward models in Langerin dynamics, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 15: ``Uncertainty Quantification'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• D. Sommer, Tensor-train kernel learning for Gaussian processes (online talk), 11th Symposium on Conformal and Probabilistic Prediction with Applications (COPA 2022) (Hybrid Event), Paper session: ``Machine Learning 1'', August 24 - 26, 2022, University of Brighton, UK, August 25, 2022.

• M. Eigel, Adaptive Galerkin FEM for non-affine linear parametric PDEs, Computational Methods in Applied Mathematics (CMAM 2022), MS06: ``Computational Stochastic PDEs'', August 29 - September 2, 2022, Technische Universität Wien, Austria, August 29, 2022.

• M. Eigel, An empirical adaptive Galerkin method for parametric PDEs, Workshop ``Adaptivity, High Dimensionality and Randomness'' (Hybrid Event), April 4 - 8, 2022, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, April 6, 2022.

• M. Eigel, Empirical adaptive Galerkin FEM for parametric PDEs, 10th International Conference on Curves and Surfaces, Minisymposium 13 ``High dimensional Approximation and PDEs'', June 20 - 24, 2022, Arcachon, France, June 23, 2022.

• R. Henrion, A turnpike property for a discrete-time linear optimal control problem with probabilistic constraints, Workshop on Optimal Control Theory, June 22 - 24, 2022, Institut National des Sciences Appliquées Rouen Normandie, France, June 24, 2022.

• R. Henrion, A turnpike property for an optimal control problem with chance constraints, PGMO DAYS 2022, Session 15F: ``New Developments in Optimal Control Theory, Part II'', November 28 - 30, 2022, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF Lab Paris-Saclay, Palaiseau, France, November 30, 2022.

• R. Henrion, Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions (online talk), Seminar on Variational Analysis and Optimization, University of Michigan, Department of Mathematics, Ann Arbor, USA, February 17, 2022.

• R. Henrion, Probabilistic constraints via spherical-radial decomposition. Part I (online talk), Seminar on Variational Analysis and Optimization, Western Michigan University, Kalamazoo, USA, February 4, 2022.

• R. Henrion, Probabilistic constraints via spherical-radial decomposition. Part II (online talk), Western Michigan University, Kalamazoo, USA, February 11, 2022.

• D. Hömberg, A phasefield approach to two-scale topology optimization, DNA Seminar (Hybrid Event), Norwegian University of Science and Technology, Department of Mathematical Sciences, Norway, March 14, 2022.

• D. Hömberg, On two-scale topology optimization (online talk), Workshop ``Practical Inverse Problems and Their Prospects'' (Online Event), March 2 - 4, 2022, Kyushu University, Japan, March 4, 2022.

• M. Landstorfer, A. Selahi, M. Heida, M. Eigel, Recovery of battery ageing dynamics with multiple timescales, MATH+-Day 2022, Technische Universität Berlin, November 18, 2022.

• R. Lasarzik, Energy-variational solutions for conservation laws, DMV Annual Meeting 2022, September 12 - 16, 2022, Freie Universität Berlin, September 14, 2022.

• R. Lasarzik, Energy-variational solutions in the context of incompressible fluid dynamics (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22), MS 47: ``Generalized Solvability Concepts for Evolutionary PDEs and their Properties'' (Online Event), March 14 - 18, 2022, Society for Industrial and Applied Mathematics, March 16, 2022.