Publications
Monographs

P.Q. Khanh, J.E. MartinezLegaz, Ch. Tammer, R. Henrion, eds., Special Issue dedicated to the 65th birthday of Alexander Kruger, 69 of Optimization, Taylor & Francis, London, 2020, (Collection Published), DOI 10.1080/02331934.2020.1815940 .
Articles in Refereed Journals

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. 
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. 
D. Hömberg, R. Lasarzik, Weak entropy solutions to a model in induction hardening, existence and weakstrong uniqueness, Mathematical Models & Methods in Applied Sciences, published online on 26.07.2021, 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. 
M.J. Arenas Jaén, D. Hömberg, R. Lasarzik, P. Mikkonen, Th. Petzold, Modelling and simulation of flame cutting for steel plates with solid phases and melting, Journal of Mathematics in Industry, 10 (2020), pp. 18/118/16, DOI 10.1186/s13362020000860 .
Abstract
The goal of this work is to describe in detail a quasistationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3Dmodel similar to previous works from Thiebaud [1] and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed. 
M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, SIAM Journal on Control and Optimization, 58 (2020), pp. 22882311, DOI 10.1137/19M1269944 .
Abstract
In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented. 
M.J. Cánovas, M.J. Gisbert, R. Henrion, J. Parra, Lipschitz lower semicontinuity moduli for linear inequality systems, Journal of Mathematical Analysis and Applications, 2 (2020), pp. 124313/1124313/21, DOI 10.1016/j.jmaa.2020.124313 .
Abstract
The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, righthand side and lefthand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuitystar as an intermediate notion between the Lipschitz lower semicontinuity and the wellknown Aubin property. We provide explicit pointbased formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings. 
T. González Grandón, R. Henrion, P. PérezAros, Dynamic probabilistic constraints under continuous random distributions, Mathematical Programming. A Publication of the Mathematical Programming Society, published online on 13.11.2020, DOI 10.1007/s1010702001593z .
Abstract
The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful twostage model with decision rules from L^{ 2} is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented. 
D. Adelhütte, D. Assmann, T. González Grandón, M. Gugat, H. Heitsch, R. Henrion, F. Liers, S. Nitsche, R. Schultz, M. Stingl, D. Wintergerst, Joint model of probabilisticrobust (probust) constraints with application to gas network optimization, Vietnam Journal of Mathematics, published online on 10.11.2020.
Abstract
Optimization problems under uncertain conditions abound in many reallife applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability. 
J.I. Asperheim, P. Das, B. Grande, D. Hömberg, Th. Petzold, Threedimensional numerical study of heat affected zone in induction welding of tubes, COMPEL. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. Emerald, Bradford, West Yorkshire. English, English abstracts., 39 (2020), pp. 213219, DOI 10.1108/COMPEL0620190238 .

F. Auricchio, E. Bonetti, M. Carraturo, D. Hömberg, A. Reali, E. Rocca, A phasefieldbased gradedmaterial topology optimization with stress constraint, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 14611483, DOI 10.1142/S0218202520500281 .
Abstract
In this paper a phasefield approach for structural topology optimization for a 3Dprinting process which includes stress constraint and potentially multiple materials or multiscales is analyzed. First order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a twodimensional cantilever beam problem. Finally, a possible workflow to obtain a 3Dprinted object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer. 
M. Drieschner, M. Eigel, R. Gruhlke, D. Hömberg, Y. Petryna, Comparison of various uncertainty models with experimental investigations regarding the failure of plates with holes, Reliability Engineering and System Safety, 203 (2020), pp. 107106/1107106/12, DOI 10.1016/j.ress.2020.107106 .
Abstract
Unavoidable uncertainties due to natural variability, inaccuracies, imperfections or lack of knowledge are always present in real world problems. To take them into account within a numerical simulation, the probability, possibility or fuzzy set theory as well as a combination of these are potentially usable for the description and quantification of uncertainties. In this work, different monomorphic and polymorphic uncertainty models are applied on linear elastic structures with nonperiodic perforations in order to analyze the individual usefulness and expressiveness. The first principal stress is used as an indicator for structural failure which is evaluated and classified. In addition to classical sampling methods, a surrogate model based on artificial neural networks is presented. With regard to accuracy, efficiency and resulting numerical predictions, all methods are compared and assessed with respect to the added value. Real experiments of perforated plates under uniaxial tension are validated with the help of the different uncertainty models. 
M. Eigel, R. Gruhlke, A local hybrid surrogatebased finite element tearing interconnecting dualprimal method for nonsmooth random partial differential equations, International Journal for Numerical Methods in Engineering, 122 (2021), pp. 10011030 (published online in December 2020), DOI 10.1002/nme.6571 .
Abstract
A domain decomposition approach exploiting the localization of random parameters in highdimensional random PDEs is presented. For high efficiency, surrogate models in multielement representations are computed locally when possible. This makes use of a stochastic Galerkin FETIDP formulation of the underlying problem with localized representations of involved input random fields. The local parameter space associated to a subdomain is explored by a subdivision into regions where the parametric surrogate accuracy can be trusted and where instead Monte Carlo sampling has to be employed. A heuristic adaptive algorithm carries out a problemdependent hp refinement in a stochastic multielement sense, enlarging the trusted surrogate region in local parametric space as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration in the involved surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, e.g. in a multiphysics setting, or when the KarhunenLoeve expansion of a random field can be localized. The efficiency of this hybrid technique is demonstrated with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. 
M. Eigel, M. Marschall, M. Multerer, An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains, SIAM/ASA Journal on Uncertainty Quantification, 8 (2020), pp. 11891214, DOI 10.1137/19M1246080 .
Abstract
A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding KarhunenLoeve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined highdimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problemdependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems. 
M. Eigel, M. Marschall, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations, Numerische Mathematik, 145 (2020), pp. 655692, DOI 10.1007/s00211020011231 .
Abstract
Stochastic Galerkin methods for nonaffine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problemadapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residualbased a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.
Preprints, Reports, Technical Reports

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. 
H. Berthold, H. Heitsch, R. Henrion, J. Schwientek, On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints, Preprint no. 2835, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2835 .
Abstract, PDF (1202 kByte)
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. 
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. 
M. Branda, R. Henrion, M. Pištěk, Value at risk approach to producer's best response in electricity market with uncertain demand, Preprint no. 2831, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2831 .
Abstract, PDF (348 kByte)
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. 
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, Preprint no. 2828, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2828 .
Abstract, PDF (596 kByte)
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. 
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. 
G. Thiele, A. Fey, D. Sommer, J. Krüger, System identification of a hysteresiscontrolled pump system using SINDy, Preprint no. 2794, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2794 .
Abstract, PDF (3533 kByte)
Hysteresiscontrolled devices are widely used in industrial applications. For example, cooling devices usually contain a twopoint controller, resulting in a nonlinear hybrid system with two discrete states. Dynamic models of systems are essential for optimizing such industrial supply technology. However, conventional system identification approaches can hardly handle hysteresiscontrolled devices. Thus, the new identification method Sparse Identification of Nonlinear Dynamics (SINDy) is extended to consider hybrid systems. SINDy composes models from basis functions out of a customized library in a datadriven manner. For modeling systems that behave dependent on their own past as in the case of natural hysteresis, Ferenc Preisach introduced the relay hysteron as an elementary mathematical description. In this new method (SINDyHybrid), tailored basis functions in form of relay hysterons are added to the library which is used by SINDy. Experiments with a hysteresis controlled water basin show that this approach correctly identifies state transitions of hybrid systems and also succeeds in modeling the dynamics of the discrete system states. A novel proximity hysteron achieves the robustness of this method. The impacts of the sampling rate and the signal noise ratio of the measurement data are examined accordingly. 
T. González Grandón, R. Henrion, P. PérezAros, Dynamic probabilistic constraints under continuous random distributions, Preprint no. 2783, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2783 .
Abstract, PDF (337 kByte)
The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful twostage model with decision rules from L^{ 2} is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented. 
M. Eigel, O. Ernst, B. Sprungk, L. Tamellini, On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion, Preprint no. 2753, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2753 .
Abstract, PDF (325 kByte)
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finitedimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residualbased 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. 
H. Heitsch, R. Henrion, An enumerative formula for the spherical cap discrepancy, Preprint no. 2744, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2744 .
Abstract, PDF (565 kByte)
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. 
R. Lasarzik, Analysis of a thermodynamically consistent NavierStokesCahnHilliard model, Preprint no. 2739, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2739 .
Abstract, PDF (305 kByte)
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. 
G. Hu, A. Rathsfeld, Radiation conditions for the Helmholtz equation in a half plane filled by inhomogeneous periodic material, Preprint no. 2726, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2726 .
Abstract, PDF (439 kByte)
In this paper we consider timeharmonic acoustic wave propagation in a halfplane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a onedimensional SturmLiouville eigenvalue problem with a complexvalued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method. 
L. Baňas, R. Lasarzik, A. Prohl, Numerical analysis for nematic electrolytes, Preprint no. 2717, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2717 .
Abstract, PDF (8587 kByte)
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. 
M. Eigel, P. Trunschke, R. Schneider, Convergence bounds for empirical nonlinear leastsquares, Preprint no. 2714, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2714 .
Abstract, PDF (4747 kByte)
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.
Talks, Poster

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, 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.

D. Sommer, tba, Leibniz MMS Summer School 2021 ``Mathematical Methods for Machine Learning'', August 23  27, 2021, Schloss Dagstuhl, LeibnizZentrum für Informatik GmbH, Wadern, August 23, 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, 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, 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, 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.

D. Sommer, A dynamic programming approach for robust receding horizon control in continuous systems (online talk), ECMI Webinar ``Math for Industry 4.0  Models, Methods and Big Data'' (Online Event), December 2  3, 2020, WIAS Berlin, December 3, 2020.

M. EbelingRump, Topology optimization under local volume constraints for improved buckling behavior (online talk), ECMI Webinar ``Math for Industry 4.0  Models, Methods and Big Data'' (Online Event), December 2  3, 2020, December 3, 2020.

R. Henrion, W. Ackooij, Analysis of a twostage probabilistic programming model, PGMO Days 2020 (Online Event), Palaiseau, France, December 1, 2020.

D. Hömberg, Maths for the digital factory (online talk), Workshop on Industrial Mathematics and Computer Science (Online Event), University of Craiova, Romania, October 31, 2020.

R. Lasarzik, Dissipative solutions in the context of the numerical approximation of nematic electrolytes (online talk), Oberseminar Numerik, Universität Bielefeld, Fakultät für Mathematik, June 23, 2020.
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