Publikationen

Artikel in Referierten Journalen

  • 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, published online on 13.05.2022, 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.

  • 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, 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, P. Trunschke, R. Schneider, 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.

  • H. Heitsch, R. Henrion, An enumerative formula for the spherical cap discrepancy, Journal of Computational and Applied Mathematics, 390 (2021), pp. 113409/1--113409/14, DOI 10.1016/j.cam.2021.113409 .
    Abstract
    The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.

  • L. Baňas, R. Lasarzik, A. Prohl, Numerical analysis for nematic electrolytes, IMA Journal of Numerical Analysis, 41 (2021), pp. 2186--2254, DOI 10.1093/imanum/draa082 .
    Abstract
    We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

  • H. Berthold, H. Heitsch, R. Henrion, J. Schwientek, On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints, Mathematical Methods of Operations Research, 96 (2022), pp. 1--37 (published online on 14.12.2021), DOI 10.1007/s00186-021-00764-8 .
    Abstract
    We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

  • G. Hu, A. Rathsfeld, W. Lu, Time-harmonic acoustic scattering from locally-perturbed periodic curves, SIAM Journal on Applied Mathematics, 81 (2021), pp. 2569--2595, DOI 10.1137/19M1301679 .
    Abstract
    We prove well-posedness for the time-harmonic acoustic scattering of plane waves from locally perturbed periodic surfaces in two dimensions under homogeneous Dirichlet boundary conditions. This covers sound-soft acoustic as well as perfectly conducting, TE polarized electromagnetic boundary value problems. Our arguments are based on a variational method in a truncated bounded domain coupled with a boundary integral representation. If the quasi-periodic Green's function to the unperturbed periodic scattering problem is calculated efficiently, then the variational approach can be used for a numerical scheme based on coupling finite elements with a boundary element algorithm.
    Even for a general 2D rough-surface problem, it turns out that the Green's function defined with the radiation condition ASR satisfies the Sommerfeld radiation condition over the half plane. Based on this result, for a local perturbation of a periodic surface, the scattered wave of an incoming plane wave is the sum of the scattered wave for the unperturbed periodic surface plus an additional scattered wave satisfying Sommerfeld's condition on the half plane. Whereas the scattered wave for the unperturbed periodic surface has a far field consisting of a finite number of propagating plane waves, the additional field contributes to the far field by a far-field pattern defined in the half-plane directions similarly to the pattern known for bounded obstacles.

  • W.M. Klesse, A. Rathsfeld, C. Gross, E. Malguth, O. Skibitzki, L. Zealouk, Fast scatterometric measurement of periodic surface structures in plasma-etching processes, Measurement, 170 (2021), pp. 108721/1--108721/12, DOI 10.1016/j.measurement.2020.108721 .
    Abstract
    To satisfy the continuous demand of ever smaller feature sizes, plasma etching technologies in microelectronics processing enable the fabrication of device structures with dimensions in the nanometer range. In a typical plasma etching system a plasma phase of a selected etching gas is activated, thereby generating highly energetic and reactive gas species which ultimately etch the substrate surface. Such dry etching processes are highly complex and require careful adjustment of many process parameters to meet the high technology requirements on the structure geometry.
    In this context, real-time access of the structure's dimensions during the actual plasma process would be of great benefit by providing full dimension control and film integrity in real-time. In this paper, we evaluate the feasibility of reconstructing the etched dimensions with nanometer precision from reflectivity spectra of the etched surface, which are measured in real-time throughout the entire etch process. We develop and test a novel and fast reconstruction algorithm, using experimental reflection spectra taken about every second during the etch process of a periodic 2D model structure etched into a silicon substrate. Unfortunately, the numerical simulation of the reflectivity by Maxwell solvers is time consuming since it requires separate time-harmonic computations for each wavelength of the spectrum. To reduce the computing time, we propose that a library of spectra should be generated before the etching process. Each spectrum should correspond to a vector of geometry parameters s.t. the vector components scan the possible range of parameter values for the geometrical dimensions. We demonstrate that by replacing the numerically simulated spectra in the reconstruction algorithm by spectra interpolated from the library, it is possible to compute the geometry parameters in times less than a second. Finally, to also reduce memory size and computing time for the library, we reduce the scanning of the parameter values to a sparse grid.

  • M. Ebeling-Rump, D. Hömberg, R. Lasarzik, Th. Petzold, Topology optimization subject to additive manufacturing constraints, Journal of Mathematics in Industry, 11 (2021), pp. 1--19, DOI 10.1186/s13362-021-00115-6 .
    Abstract
    In Topology Optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg-Landau term. During 3D Printing overhangs lead to instabilities, which have only been tackled unsatisfactorily. The novel idea is to incorporate an Additive Manufacturing Constraint into the phase field method. A rigorous analysis proves the existence of a solution and leads to first order necessary optimality conditions. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the Additive Manufacturing Constraint brings about support structures, which can be fine tuned according to engineering demands. Stability during 3D Printing is assured, which solves a common Additive Manufacturing problem.

  • D. Hömberg, R. Lasarzik, Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness, Mathematical Models & Methods in Applied Sciences, 31 (2021), pp. 1867--1918, DOI 10.1142/S021820252150041X .
    Abstract
    In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions.

  • R. Lasarzik, Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model, Nonlinear Analysis. An International Mathematical Journal, 213 (2021), pp. 112526/1--112526/33, DOI 10.1016/j.na.2021.112526 .
    Abstract
    In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.

  • R. Lasarzik, Maximally dissipative solutions for incompressible fluid dynamics, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 73 (2022), pp. 1/1--1/21 (published online on 11.11.2021), DOI 10.1007/s00033-021-01628-1 .
    Abstract
    We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.

Beiträge zu Sammelwerken

  • 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

  • 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)
    oduce 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. Lasarzik, M.E.V. Reiter, Analysis and numerical approximation of energy-variational solutions to the Ericksen--Leslie equations, Preprint no. 2966, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2966 .
    Abstract, PDF (8159 kByte)
    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.

  • M. Gugat, H. Heitsch, R. Henrion, A turnpike property for optimal control problems with dynamic probabilistic constraints, Preprint no. 2941, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2941 .
    Abstract, PDF (354 kByte)
    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.

  • 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, Preprint no. 2928, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2928 .
    Abstract, PDF (9378 kByte)
    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.

  • 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.

  • D. Hömberg, R. Lasarzik, L. Plato, On the existence of generalized solutions to a spatio-temporal predator-prey system, Preprint no. 2925, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2925 .
    Abstract, PDF (1235 kByte)
    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.

  • M. Eigel, R. Gruhlke, D. Moser, Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains, Preprint no. 2907, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2907 .
    Abstract, PDF (2369 kByte)
    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, N. Farchmin, S. Heidenreich, P. Trunschke, Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs, Preprint no. 2897, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2897 .
    Abstract, PDF (507 kByte)
    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, R. Schneider, D. Sommer, Dynamical low-rank approximations of solutions to the Hamilton--Jacobi--Bellman equation, Preprint no. 2896, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2896 .
    Abstract, PDF (399 kByte)
    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.

  • R. Henrion, A. Jourani, B.S. Mordukhovich, Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions, Preprint no. 2892, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2892 .
    Abstract, PDF (366 kByte)
    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.

  • 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, Preprint no. 2866, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2866 .
    Abstract, PDF (3420 kByte)
    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. Eigel, N. Farchmin, S. Heidenreich, P. Trunschke, Efficient approximation of high-dimensional exponentials by tensor networks, Preprint no. 2844, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2844 .
    Abstract, PDF (349 kByte)
    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. Lasarzik, On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics, Preprint no. 2834, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2834 .
    Abstract, PDF (264 kByte)
    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.

  • CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing high-dimensional Bermudan options with hierarchical tensor formats, Preprint no. 2821, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2821 .
    Abstract, PDF (321 kByte)
    An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the “curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.

Vorträge, Poster

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

  • J. Schütte, tba, Annual Meeting of SPP 2298, November 20 - 23, 2022, Evangelische Akademie, Tutzing.

  • 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. 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, Dynamical low rank approximation of the Kolmogorov backward equation for posterior estimation in Bayesian 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.

  • 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.

  • 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, Rouen, 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 29, 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.

  • R. Henrion, Probabilistic constraints via spherical-radial decomposition. Part I (online talk), 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.

  • 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.

  • L. Plato, R. Lasarzik, D. Hömberg, E. Emmrich, Nonlinear electrokinetics in anisotropic microfluids --- Analysis, simulation, and optimal control, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

  • R. Gruhlke, Annual report 2021 -- MuScaBlaDes (subproject 4 within SPP1886) (online talk), Jahrestreffen des SPP 1886 (Online Event), October 25 - 26, 2021, October 25, 2021.

  • H. Heitsch, An algorithmic approach for solving optimization problems with probabilistic/robust (probust) constraints, Workshop ``Applications of Semi-Infinite Optimization'' (Online Event), May 20 - 21, 2021, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, May 21, 2021.

  • D. Sommer, Feedforward neural networks for regression problems, Leibniz MMS Summer School 2021 ``Mathematical Methods for Machine Learning'', August 23 - 27, 2021, Schloss Dagstuhl, Leibniz-Zentrum für Informatik GmbH, Wadern, August 23, 2021.

  • D. Sommer, Robust nonlinear model predictive control using tensor networks (online talk), European Conference on Mathematics for Industry (ECMI2021), MS23: ``Data-Driven Optimization'' (Online Event), April 13 - 15, 2021, Bergische Universität Wuppertal, April 14, 2021.

  • M. Ebeling-Rump, Topology optimization subject to a local volume constraint (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2020@21), Section S19 ``Optimization of Differential Equations'' (Online Event), March 15 - 19, 2021, Universität Kassel, March 16, 2021.

  • M. Ebeling-Rump, Topology optimization subject to a local volume constraint (online talk), European Conference on Mathematics for Industry (ECMI2021), MS07: ``Maths for the Digital Factory'' (Online Event), April 13 - 15, 2021, Bergische Universität Wuppertal, April 15, 2021.

  • M. Eigel, A neural multilevel method for high-dimensional parametric PDEs, Thirty-fifth Conference on Neural Information Processing Systems (NeurIPS 2021) (Online Event), December 6 - 14, 2021.

  • M. Eigel, An adaptive tensor reconstruction for Bayesian inversion (online talk), School for Simulation and Data Science (SSD) Seminar, RWTH Aachen, IRTG Modern Inverse Problems, July 5, 2021.

  • M. Eigel, Introduction to machine learning: Neural networks, Leibniz MMS Summer School 2021 ``Mathematical Methods for Machine Learning'', August 23 - 27, 2021, Schloss Dagstuhl, Leibniz-Zentrum für Informatik GmbH, Wadern, August 23, 2021.

  • R. Henrion, Adaptive grid refinement for optimization problems with probabilistic/robust (probust) constraints, PGMO DAYS 2021, Session 12E ``Stochastic Optimization I'', November 30 - December 1, 2021, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF Lab Paris-Saclay, Palaiseau, France, December 1, 2021.

  • R. Henrion, Contraintes en probabilité au-delà de la recherche opérationnelle (online talk), 13e Journée Normandie-Mathématique (Hybrid Event), Rouen, France, June 24, 2021.

  • R. Henrion, Dealing with probust constraints in stochastic optimization, Workshop ``Applications of Semi-Infinite Optimization'' (Online Event), May 20 - 21, 2021, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, May 21, 2021.

  • D. Hömberg, Industry 4.0 -- Mathematical concepts and new challenges (online talk), International Conference on Direct Digital Manufacturing and Polymers (ICDDMAP 2021) (Online Event), May 20 - 22, 2021, Polytechnic of Leiria, Portugal, May 21, 2021.

  • D. Hömberg, Mathematics for steel production and manufacturing (online talk), Cardiff University, School of Mathematics, UK, March 2, 2021.

  • D. Hömberg, Modelling and simulation of high-frequency induction welding (online talk), European Conference on Mathematics for Industry (ECMI2021), MS08: ``Modelling, Simulation and Optimization in Electrical Engineering'' (Online Event), April 13 - 15, 2021, Bergische Universität Wuppertal, April 13, 2021.

  • M. Landstorfer, M. Eigel, M. Heida, A. Selahi, Recovery of battery ageing dynamics with multiple timescales (online poster), MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

  • R. Lasarzik, Energy-variational solutions for incompressible fluid dynamics, Oberseminar Angewandte Analysis, Universität Ulm, Institut für Angewandte Analysis, October 25, 2021.

  • R. Lasarzik, Energy-variational solutions for incompressible fluid dynamics, Technische Universität Berlin, Institut für Mathematik, November 8, 2021.