Publications
Articles in Refereed Journals

H. Heitsch, R. Henrion, Th. Kleinert, M. Schmidt, On convex lowerlevel blackbox 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/s1089802201161z .
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 blackbox constraints in the lower level. For this setup, we develop a cuttingplane algorithm that computes approximate bilevelfeasible 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 blackbox setting studied before. For the applied model, we use further problemspecific 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 valueatrisk 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. Eigel, R. Gruhlke, M. Marschall, Lowrank tensor reconstruction of concentrated densities with application to Bayesian inversion, Statistics and Computing, 32 (2022), pp. 27/127/27, DOI 10.1007/s11222022100871 .
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 lowrank 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 KullbackLeibler 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, M. Haase, J. Neumann, Topology optimisation under uncertainties with neural networks, Algorithms, 15 (2022), pp. 241/1241/34, DOI https://doi.org/10.3390/a15070241 .

M. Eigel, P. Trunschke, R. Schneider, Convergence bounds for empirical nonlinear leastsquares, ESAIM: Mathematical Modelling and Numerical Analysis, 56 (2022), pp. 79104, 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 wellknown 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. 
R. Lasarzik, E. Rocca, G. Schimperna, Weak solutions and weakstrong uniqueness for a thermodynamically consistent phasefield model, Rendiconti Lincei  Matematica e Applicazioni, published online on 28.07.2022, DOI 10.4171/RLM/970 .
Abstract
In this paper we prove the existence of weak solutions for a thermodynamically consistent phasefield 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 localintime strong solutions and, finally, we show weakstrong 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. 113409/1113409/14, 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. 21862254, 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, structureinheriting spacetime 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. 137 (published online on 14.12.2021), DOI 10.1007/s00186021007648 .
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, Timeharmonic acoustic scattering from locallyperturbed periodic curves, SIAM Journal on Applied Mathematics, 81 (2021), pp. 25692595, DOI 10.1137/19M1301679 .
Abstract
We prove wellposedness for the timeharmonic acoustic scattering of plane waves from locally perturbed periodic surfaces in two dimensions under homogeneous Dirichlet boundary conditions. This covers soundsoft 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 quasiperiodic 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 roughsurface 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 farfield pattern defined in the halfplane 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 plasmaetching processes, Measurement, 170 (2021), pp. 108721/1108721/12, 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, realtime 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 realtime. 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 realtime 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 timeharmonic 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. EbelingRump, D. Hömberg, R. Lasarzik, Th. Petzold, Topology optimization subject to additive manufacturing constraints, Journal of Mathematics in Industry, 11 (2021), pp. 119, DOI 10.1186/s13362021001156 .
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 GinzburgLandau 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 AllenCahn 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 weakstrong uniqueness, Mathematical Models & Methods in Applied Sciences, 31 (2021), pp. 18671918, 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 weakstrong 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 NavierStokesCahnHilliard model, Nonlinear Analysis. An International Mathematical Journal, 213 (2021), pp. 112526/1112526/33, DOI 10.1016/j.na.2021.112526 .
Abstract
In this paper, existence of generalized solutions to a thermodynamically consistent NavierStokesCahnHilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measurevalued and dissipative solutions. The measurevalued 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 measurevalued 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 weakstrong 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. 1/11/21 (published online on 11.11.2021), DOI 10.1007/s00033021016281 .
Abstract
We introduce the new concept of maximal dissipative solutions for the NavierStokes 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.
Preprints, Reports, Technical Reports

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 timediscrete 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 wellknown 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 fiberreinforced 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 onescale 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, Lowrank 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 multielement 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 noncontinuous and noninjective 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 lowrank 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 nonperiodic 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 spatiotemporal predatorprey 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 nonlinear partial differential equations describing the interaction of a predatorprey 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 weakstrong uniqueness of these generalized solutions and the existence of strong solutions at least locallyintime for space dimension two and three. 
M. EbelingRump, D. Hömberg, R. Lasarzik, Twoscale topology optimization with heterogeneous mesostructures based on a local volume constraint, Preprint no. 2908, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2908 .
Abstract, PDF (17 MByte)
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 phasefield 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, 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 lowrank 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 starshaped inclusions is demonstrated. 
TH. Eiter, K. Hopf, R. Lasarzik, Weakstrong uniqueness and energyvariational solutions for a class of viscoelastoplastic fluid models, Preprint no. 2904, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2904 .
Abstract, PDF (345 kByte)
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 ZarembaJaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show shorttime existence of strong solutions as well as their uniqueness in a class of LerayHopf type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The globalintime 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 energyvariational solutions, which is based on an inequality for the relative energy. We derive general properties of energyvariational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion. 
M. Eigel, N. Farchmin, S. Heidenreich, P. Trunschke, Adaptive nonintrusive reconstruction of solutions to highdimensional 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 nonintrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the nonintrusive character of a randomized leastsquares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasioptimal lowrank 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 nonintrusive adaptive algorithm, showing bestinclass performance. 
M. Eigel, R. Schneider, D. Sommer, Dynamical lowrank approximations of solutions to the HamiltonJacobiBellman 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 lowrank tensor train (TT) decompositions. The approach is based on the DiracFrenkel variationalprinciple with the modification that the optimisation uses an empirical risk. Compared to currentstateoftheart 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 finitedimensional 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 highaccuracy 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 perfectlymatchedlayer (PML) method for wave scattering in a half space of homogeneous medium bounded by a twodimensional, perfectly conducting, and locally defected periodic surface, and develops a highaccuracy boundaryintegralequation (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 semiwaveguide regions, separated by two vertical line segments. In both semiwaveguides, we prove the wellposedness of an associated scattering problem so as to well define a NeumanntoDirichlet (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 semiwaveguides, both NtD operators turn out to be closely related to a Neumannmarching operator, governed by a nonlinear Riccati equation. It is proved that the Neumannmarching 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 highaccuracy PMLbased 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 highdimensional 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 highdimensional function h is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of lognormal random fields or the evaluation of Bayesian posterior measures. Usually, these highdimensional 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 PetrovGalerkin 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 lognormal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent lowrank 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 highdimensional 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 energyvariational solutions for the NavierStokes 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 energyvariational solutions and thus weak solutions in any space dimension for the NavierStokes 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 wellposedness for these equations. 
CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing highdimensional 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 leastsquares approach as well as the dual martingale method, both using highdimensional 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.
Talks, Poster

L. Plato, Generalized solutions in the context of a nonlocal predetorprey 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, RheinischWestfälische Technische Hochschule Aachen, August 16, 2022.

J. Schütte, Adaptive neural tensor networks for parametric PDEs, Workshop on the Approximation of Solutions of HighDimensional 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.

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, RheinischWestfälische 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, RheinischWestfälische 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, MASCOTNUM 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, RheinischWestfälische Technische Hochschule Aachen, August 16, 2022.

M. Eigel, Adaptive Galerkin FEM for nonaffine 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 discretetime 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, 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 sphericalradial decomposition. Part I (online talk), Western Michigan University, Kalamazoo, USA, February 4, 2022.

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

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

D. Hömberg, On twoscale 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, Energyvariational solutions for conservation laws, DMV Annual Meeting 2022, September 12  16, 2022, Freie Universität Berlin, September 14, 2022.

R. Lasarzik, Energyvariational 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 SemiInfinite Optimization'' (Online Event), May 20  21, 2021, FraunhoferInstitut 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, LeibnizZentrum 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: ``DataDriven Optimization'' (Online Event), April 13  15, 2021, Bergische Universität Wuppertal, April 14, 2021.

M. EbelingRump, 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. EbelingRump, 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 highdimensional parametric PDEs, Thirtyfifth 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, LeibnizZentrum 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 ParisSaclay, Palaiseau, France, December 1, 2021.

R. Henrion, Contraintes en probabilité audelà de la recherche opérationnelle (online talk), 13e Journée NormandieMathématique (Hybrid Event), Rouen, France, June 24, 2021.

R. Henrion, Dealing with probust constraints in stochastic optimization, Workshop ``Applications of SemiInfinite Optimization'' (Online Event), May 20  21, 2021, FraunhoferInstitut 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 highfrequency 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, Energyvariational solutions for incompressible fluid dynamics, Oberseminar Angewandte Analysis, Universität Ulm, Institut für Angewandte Analysis, October 25, 2021.

R. Lasarzik, Energyvariational solutions for incompressible fluid dynamics, Technische Universität Berlin, Institut für Mathematik, November 8, 2021.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations