Publikationen
Monografien

R. Henrion, Chapter 2: Calmness as a Constraint Qualification for MStationarity Conditions in MPECs, in: Generalized Nash Equilibrium Problems, Bilevel Programming and MPEC, D. Aussel, C.S. Lalitha, eds., Forum for Interdisciplinary Mathematics, Springer, Singapore, 2018, pp. 2141, (Chapter Published), DOI 10.1007/9789811047749 .
Artikel in Referierten Journalen

W. VAN Ackooij, R. Henrion, P. PérezAros, Generalized gradients for probabilistic/robust (probust) constraints, Optimization. A Journal of Mathematical Programming and Operations Research, published online on 14.02.2019, DOI 10.1080/02331934.2019.1576670 .
Abstract
Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semiinfinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data. 
C. Brée, D. Gailevičius, V. Purlys, G.G. Werner, K. Staliunas, A. Rathsfeld, G. Schmidt, M. Radziunas, Chirped photonic crystal for spatially filtered optical feedback to a broadarea laser, Journal of Optics, 20 (2018), pp. 095804/1095804/7, DOI 10.1088/20408986/aada98 .
Abstract
We derive and analyze an efficient model for reinjection of spatially filtered optical feedback from an external resonator to a broad area, edge emitting semiconductor laser diode. Spatial filtering is achieved by a chirped photonic crystal, with variable periodicity along the optical axis and negligible resonant backscattering. The optimal chirp is obtained from a genetic algorithm, which yields solutions that are robust against perturbations. Extensive numerical simulations of the composite system with our optoelectronic solver indicate that spatially filtered reinjection enhances lowerorder transversal optical modes in the laser diode and, consequently, improves the spatial beam quality. 
M. Eigel, J. Neumann, R. Schneider, S. Wolf, Risk averse stochastic structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 334 (2018), pp. 470482, DOI 10.1016/j.cma.2018.02.003 .
Abstract
A novel approach for riskaverse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a highdimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common riskaware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such highdimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach. 
M. Eigel, M. Marschall, R. Schneider, Samplingfree Bayesian inversion with adaptive hierarchical tensor representations, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 035010/1035010/29, DOI 10.1088/13616420/aaa998 .
Abstract
The statistical Bayesian approach is a natural setting to resolve the illposedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a samplingfree approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed samplingfree approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent highdimensional quadrature of the loglikelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the lowrank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results. 
L. Adam, M. Branda, H. Heitsch, R. Henrion, Solving joint chance constrained problems using regularization and Benders' decomposition, Annals of Operations Research, pp. published online on 08.11.2018, urlhttps://doi.org/10.1007/s1047901830919, DOI 10.1007/s1047901830919 .
Abstract
In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution. 
E. Emmrich, S.H.L. Klapp, R. Lasarzik, Nonstationary models for liquid crystals: A fresh mathematical perspective, Journal of NonNewtonian Fluid Mechanics, 259 (2018), pp. 3247, DOI 10.1016/j.jnnfm.2018.05.003 .

E. Emmrich, R. Lasarzik, Existence of weak solutions to the EricksenLeslie model for a general class of free energies, Mathematical Methods in the Applied Sciences, 41 (2018), pp. 64926518.

E. Emmrich, R. Lasarzik, Weakstrong uniqueness for the general EricksenLeslie system in three dimensions, Discrete and Continuous Dynamical Systems, 38 (2018), pp. 46174635, DOI 10.3934/dcds.2018202 .

A. Hantoute, R. Henrion, P. PérezAros, Subdifferential characterization of probability functions under Gaussian distribution, Mathematical Programming. A Publication of the Mathematical Programming Society, 174 (2019), pp. 167194 (published online on 29.01.2018), DOI 10.1007/s1010701812379 .
Abstract
Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth. This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae of such functions in the case of Gaussian distributions for possibly infinitedimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the sphericradial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other. By successively adding additional hypotheses, conditions are satisfied under which the probability function is locally Lipschitzian or even differentiable. 
M. Eigel, J. Neumann, R. Schneider, S. Wolf, Nonintrusive tensor reconstruction for high dimensional random PDEs, Computational Methods in Applied Mathematics, 19 (2019), pp. 3953 (published online on 25.07.2018), DOI 10.1515/cmam20180028 .
Abstract
This paper examines a completely nonintrusive, samplebased method for the computation of functional lowrank solutions of high dimensional parametric random PDEs which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel blackbox rankadapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling. 
R. Henrion, W. Römisch, Problembased optimal scenario generation and reduction in stochastic programming, Mathematical Programming. A Publication of the Mathematical Programming Society, pp. published online on 04.10.2018, urlhttps://doi.org/10.1007/s1010701813376, DOI 10.1007/s1010701813376 .
Abstract
Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear twostage stochastic programs we show that the problembased approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semiinfinite program. We show that the latter is convex if either righthand sides or costs are random and can be transformed into a semiinfinite program in a number of cases. We also consider problembased optimal scenario reduction for twostage models and optimal scenario generation for chance constrained programs. Finally, we discuss problembased scenario generation for the classical newsvendor problem. 
R. Lasarzik, Dissipative solution to the EricksenLeslie system equipped with the OseenFrank energy, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, pp. published online on 29.11.2018, urlhttps://doi.org/10.1007/s0003301810533, DOI 10.1007/s0003301810533 .
Abstract
We analyze the EricksenLeslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measurevalued solutions to the considered system and showed global existence as well as weakstrong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weakstrong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the longtime behavior and show that all solutions converge for the large time limit to a certain steady state.
Beiträge zu Sammelwerken

M.J. Cánovas, R. Henrion, M.A. López, J. Parra, Indexation strategies and calmness constants for uncertain linear inequality systems, in: The Mathematics of the Uncertain, E. Gil, E. Gil, J. Gil, M.Á. Gil, eds., 142 of Studies in Systems, Decision and Control, Springer, 2018, pp. 831843.
Preprints, Reports, Technical Reports

M. Eigel, L. Grasedyck, R. Gruhlke, D. Moser, Low rank surrogates for polymorphic fields with application to fuzzystochastic partial differential equations, Preprint no. 2580, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2580 .
Abstract, PDF (1235 kByte)
We consider a general form of fuzzystochastic PDEs depending on the interaction of probabilistic and nonprobabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for realworld applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and welldefinedness of polymorphic PDEs in appropriate function spaces. The fuzzystochastic dependence is described in a highdimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed lowrank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a nonintrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with KarhunenLoeve coefficient field which is generalized by a fuzzy correlation length. 
M. Drieschner, M. Eigel, R. Gruhlke, D. Hömberg, Y. Petryna, Comparison of monomorphic and polymorphic approaches for uncertainty quantification with experimental investigations, Preprint no. 2579, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2579 .
Abstract, PDF (6838 kByte)
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. 
J. Elschner, G. Hu, Inverse elastic scattering from rigid scatterers with a single incoming wave, Preprint no. 2571, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2571 .
Abstract, PDF (383 kByte)
The first part of this paper is concerned with uniqueness to inverse timeharmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the farfield pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single farfield pattern. 
E. Emmrich, R. Lasarzik, Existence of weak solutions to a dynamic model for smecticA liquid crystals under undulations, Preprint no. 2567, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2567 .
Abstract, PDF (381 kByte)
A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smecticA liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the threedimensional case. 
M. Eigel, M. Marschall, M. Multerer, An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains, Preprint no. 2566, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2566 .
Abstract, PDF (5791 kByte)
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, R. Gruhlke, A hybrid FETIDP method for nonsmooth random partial differential equations, Preprint no. 2565, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2565 .
Abstract, PDF (3236 kByte)
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. 
W.M. Klesse, A. Rathsfeld, C. Gross, E. Malguth, O. Skibitzki, L. Zealouk, Fast scatterometric measurement of periodic surface structures in plasmaetching processes, Preprint no. 2564, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2564 .
Abstract, PDF (4576 kByte)
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. Carraturo, E. Rocca, E. Bonetti, D. Hömberg, A. Reali, F. Auricchio, Additive manufacturing gradedmaterial design based on phasefield and topology optimization, Preprint no. 2553, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2553 .
Abstract, PDF (1965 kByte)
In the present work we introduce a novel gradedmaterial design for additive manufacturing based on phasefield and topology optimization. The main novelty of this work comes from the introduction of an additional phasefield variable in the classical singlematerial phasefield topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Two different numerical examples are discussed, in both of them we perform sensitivity studies to asses the effects of different model parameters onto the resulting structure. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology. 
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, Preprint no. 2550, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2550 .
Abstract, PDF (8736 kByte)
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. 
M. Eigel, P. Trunschke, R. Schneider, S. Wolf, Variational Monte Carlo  Bridging concepts of machine learning and high dimensional partial differential equations, Preprint no. 2544, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2544 .
Abstract, PDF (598 kByte)
A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors. 
H. Heitsch, On probabilistic capacity maximization in a stationary gas network, Preprint no. 2540, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2540 .
Abstract, PDF (379 kByte)
The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach to assist gas network operators in managing uncertainty with respect to the demand and in exposing free network capacities while increasing reliability of transmission and supply. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving an optimization problem with joint probabilistic constraints over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing free booked capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. We are going to validate a kind of rank two constraint qualification implying the differentiability of the considered capacity problem. At the numerical side we are going to solve the problem using a projected gradient decent method, where the function and gradient evaluations of the probabilistic constraints are performed by the approach of sphericradial decomposition applied for multivariate Gaussian random variables and more general distributions. 
R. Lasarzik, Approximation and optimal control of dissipative solutions to the EricksenLeslie system, Preprint no. 2535, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2535 .
Abstract, PDF (308 kByte)
We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measurevalued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the normrestriction on the director in every step. We introduce a semidiscrete scheme and derive an approximated version of the relativeenergy inequality for solutions of this scheme. Passing to the limit in the semidiscretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semicontinuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity. 
J. Yu, D. Hömberg, Th. Petzold, S. Lu, Temporal homogenization of a nonlinear parabolic system, Preprint no. 2524, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2524 .
Abstract, PDF (982 kByte)
In this paper we develop are twoscale model for a nonlinear parabolic system. Assuming a rapidly oscillating inhomogeneity with period ε for one equation we carry out a formal periodic expansion to obtain a homogenized equation coupled to a local in time cell problem. We justify the expansion by deriving an error estimate between the original and the twoscale model and show numerical simulations, which confirm the analytically derived error estimate. 
G. Hu, A. Rathsfeld, Acoustic scattering from locally perturbed periodic surfaces, Preprint no. 2522, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2522 .
Abstract, PDF (298 kByte)
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. 
M. Eigel, M. Marschall, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations, Preprint no. 2515, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2515 .
Abstract, PDF (557 kByte)
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. 
H. Heitsch, N. Strogies, Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines, Preprint no. 2513, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2513 .
Abstract, PDF (474 kByte)
Assuming a pipewise constant structure of the friction coefficient in the modeling of natural gas transport through a passive network of pipes via semilinear systems of balance laws with associated linear coupling and boundary conditions, uncertainty in this parameter is quantified by a Markov chain Monte Carlo method. Here, information on the prior distribution is obtained from practitioners. The results are applied to the problem of validating technical feasibility under random exit demand in gas transport networks. In particular, the impact of quantified uncertainty to the probability level of technical feasible exit demand situations is studied by two example networks of small and medium size. The gas transport of the network is modeled by stationary solutions that are steady states of the time dependent semilinear problems. 
R. Lasarzik, Dissipative solution to the EricksenLeslie system equipped with the OseenFrank energy, Preprint no. 2508, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2508 .
Abstract, PDF (315 kByte)
We analyze the EricksenLeslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measurevalued solutions to the considered system and showed global existence as well as weakstrong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weakstrong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the longtime behavior and show that all solutions converge for the large time limit to a certain steady state. 
R. Lasarzik, Measurevalued solutions to the EricksenLeslie model equipped with the OseenFrank energy, Preprint no. 2476, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2476 .
Abstract, PDF (335 kByte)
In this article, we prove the existence of measurevalued solutions to the EricksenLeslie system equipped with the OseenFrank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measurevalued solutions for vanishing regularization. Additionally, it is shown that the measurevalued solution fulfills an energy inequality. 
R. Lasarzik, Weakstrong uniqueness for measurevalued solutions to the EricksenLeslie model equipped with the OseenFrank free energy, Preprint no. 2474, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2474 .
Abstract, PDF (323 kByte)
We analyze the EricksenLeslie system equipped with the OseenFrank energy in three space dimensions. Recently, the author introduced the concept of measurevalued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measurevalued solutions, which fulfill an associated energy inequality, enjoy the weakstrong uniqueness property, i.e. the measurevalued solution agrees with a strong solution if the latter exists. The weakstrong uniqueness is shown by a relative energy inequality for the associated nonconvex energy functional. 
L. Adam, M. Branda, H. Heitsch, R. Henrion, Solving joint chance constrained problems using regularization and Benders' decomposition, Preprint no. 2471, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2471 .
Abstract, PDF (236 kByte)
In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution.
Vorträge, Poster

R. Gruhlke, Polymorphic uncertainty propagation with application to failure analysis ofadhesive bonds in rotor blades, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Session ``DFGPP 1886: Polymorphic uncertainty modelling for the numerical design of structures'', February 18  22, 2019, Technische Universität Wien, Austria, February 19, 2019.

M. EbelingRump, Topology optimization subject to additive manufacturing constraints, INdAM Workshop MACH2019 ``Mathematical modeling and Analysis of degradation and restoration in Cultural Heritage'', March 25  29, 2019, Istituto Nazionale di Alta Matematica "Francesco Severi", Rome, Italy, March 26, 2019.

M. EbelingRump, Workshop on Numerical Methods for Optimal Control and Inverse Problems (OCIP 2019), March 11  13, 2019, Leibniz Supercomputing Centre (LRZ), Garching.

M. Eigel, A statistical learning approach for parametric PDEs, Workshop ``Scientific Computation using MachineLearning Algorithms'', April 25  26, 2019, University of Nottingham, UK, April 26, 2019.

R. Henrion, Robust control of a sweeping process with probabilistic endpoint constraints, Workshop ``Nonsmooth and Variational Analysis'', January 28  February 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, January 31, 2019.

D. Hömberg, From distortion compensation to 3D printing  A phase field approach to topology optimization, The Hawassa Math & Stat Conference 2019, February 11  15, 2019, Hawassa University, Ethiopia, February 12, 2019.

R. Lasarzik, Optimal control via relative energies, Workshop ``Recent Trends in Optimal Control of Partial Differential Equations'', February 25  27, 2019, Technische Universität Berlin, February 27, 2019.

R. Lasarzik, Weak entropic solutions to a model in induction hardening: Existence and weakstrong uniqueness, Workshop ``Decima Giornata di Studio Università di Pavia  Politecnico di Milano Equazioni Differenziali e Calcolo delle Variazioni'', Politecnico di Milano, Italy, February 21, 2019.

M. Marschall, Random domains in PDE problems with lowrank surrogates. Forward and backward, PhysikalischTechnische Bundesanstalt, Berlin, April 10, 2019.

M.J. Arenas Jaén, Modelling, simulation and optimization of inductive preheating for thermal cutting of steel plates, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium 27 ``MSO for Steel Production and Manufacturing'', June 18  22, 2018, Budapest, Hungary, June 19, 2018.

L. Capone, Induction hardening of cam profiles: Modeling, simulation, and optimization, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium 27 ``MSO for Steel Production and Manufacturing'', June 18  22, 2018, Budapest, Hungary, June 19, 2018.

M. Eigel, Aspects of adaptive Galerkin FE for stochastic direct and inverse problems, Workshop ``Surrogate Models for UQ in Complex Systems'' (UNQW02), February 5  9, 2018, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, February 7, 2018.

R. Gruhlke, Domain decomposition and multiscale failure analysis with polymorphic uncertainties for optimal design of rotor blades, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), March 19  23, 2018, Technische Universität München, March 22, 2018.

R. Gruhlke, Domain decomposition, random media and interaction, SPP 1886 Workshop ``Komplex C'', July 10  11, 2018, RuhrUniversität Bochum, July 10, 2018.

R. Gruhlke, Stochastic domain decomposition and application in heterogeneous material, 3rd GAMM AGUQ Workshop on Uncertainty Quantification 2018, TU Dortmund, March 12  14, 2018.

H. Heitsch, A probabilistic approach to optimization in gas transport, 2nd Conference on Mathematics of Gas Transport (MoG2), October 10  11, 2018, KonradZuseZentrum für Informationstechnik Berlin, October 10, 2018.

H. Heitsch, Stochastic optimization with probabilistic/robust (probust) constraints, 23rd International Symposium on Mathematical Programming (ISMP 2018), Cluster: Optimization under Uncertainty, July 1  6, 2018, Bordeaux, France, July 2, 2018.

I. Bremer, Using an SQPsolver for nonlinear optimization under probabilistic constraints, 100th Meeting of the GOR Working Group ``Real World Mathematical Optimization'', April 12  13, 2018, Regenstauf, April 12, 2018.

M. Eigel, R. Gruhlke, Domain decomposition for random highdimensional PDEs, Workshop ``Reducing dimensions and cost for UQ in complex systems'', Cambridge, UK, March 5  9, 2018.

M. Eigel, A statistical learning view on random PDEs, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), March 19  23, 2018, Technische Universität München, March 20, 2018.

M. Eigel, Adaptive Galerkin FEM for stochastic forward and inverse problems, Optimisation and Numerical Analysis Seminars, University of Birmingham, School of Mathematics, UK, February 15, 2018.

M. Eigel, Adaptive tensor methods for forward and inverse problems, SIAM Conference on Uncertainty Quantification (UQ18), Minisymposium 122 ``LowRank Approximations for the Forward and the Inverse Problems III'', April 16  19, 2018, Garden Grove, USA, April 19, 2018.

R. Henrion, Dynamic chance constraints under continuous random distribution, PGMODAYS 2018, Session 1E ``Stochastic optimization'', November 20  21, 2018, EDF'Lab ParisSaclay, Palaiseau, France, November 21, 2018.

R. Henrion, Dynamic chance constraints under random distributions, 23rd International Symposium on Mathematical programming (ISMP 2018), Cluster: Optimization under Uncertainty, July 1  6, 2018, Bordeaux, France, July 5, 2018.

R. Henrion, Lipschitz properties and their moduli for constraint mappings, 4th Central European SetValued and Variational Analysis Meeting (CESVVAM 2018), November 24, 2018, PhilippsUniversität Marburg, November 24, 2018.

R. Henrion, Mstationary conditions for MPECs in finite dimension (Part 1+2), SPP 1962 Summer School on Complementarity Problems in Applied Mathematics: Modeling, Analysis, and Optimization, July 30  August 1, 2018, Technische Universität Dortmund, July 31, 2018.

R. Henrion, Maximization of free capacities in gas networks with random load, Conference ``Variational Analysis  Challenges in Energy'', June 4  6, 2018, Castro Urdiales, Spain, June 4, 2018.

R. Henrion, Optimization problems under probabilistic constraints, 3rd RussianGerman Conference on MultiScale BioMathematics: Coherent Modeling of Human Body System, November 7  9, 2018, Lomonosov Moscow State University, Russian Federation, November 8, 2018.

R. Henrion, Optimization problems with probust constraints, Colloquium & International Conference on Variational Analysis and Nonsmooth Optimization, June 28  July 1, 2018, MartinLutherUniversität HalleWittenberg, Halle, June 28, 2018.

R. Henrion, Perspectives in probabilistic programming under continuous random distributions, Workshop ``New Directions in Stochastic Optimisation'', August 19  25, 2018, Mathematisches Forschungsinstitut Oberwolfach, August 20, 2018.

R. Henrion, Probabilistic programming with applications, Universidad Miguel Hernández de Elche, Instituto Centro de Investigación Operativa, Spain, September 13, 2018.

R. Henrion, Verification and comparison of the calmness of generalized equations in original and enhanced form, International Workshop on Optimization and Variational Analysis, January 10  11, 2018, Termas de Cauquenes, Chile, January 11, 2018.

D. Hömberg, European collaboration in industrial and applied mathematics, Conference ``Mathematical Modelling in Metallurgical Industry'', September 17  18, 2018, Kristiansand, Norway, September 18, 2018.

D. Hömberg, European collaboration in industrial and applied mathematics, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, May 17, 2018.

D. Hömberg, European collaboration in industrial and applied mathematics, Inderprastha Engineering College, Ghaziabad, India, January 5, 2018.

D. Hömberg, Knowledge exchange in mathematics  The european perspective, 1st IMA Conference on Knowledge Exchange in the Mathematical Sciences, December 3  4, 2018, Aston University, Birmingham, UK, December 3, 2018.

D. Hömberg, Mathematical challenges in steel manufacturing, Second International Conference on Modern Mathematical Methods and High Performance Computing in Science and Technology (M3HPCST 2018), January 4  6, 2018, Inderprastha Engineering College (IPEC), Ghaziabad, India, January 4, 2018.

D. Hömberg, Maths for digital factory, University of Southern Denmark, The Maersk McKinney Moller Institute, Odense, Denmark, October 12, 2018.

D. Hömberg, Temporal homogenization of a nonlinear parabolic system, The 20th European Conference on Mathematics for Industry (ECMI 2018), Minisymposium 15 ``Electromagnetic Problems Arising in Industry: Modelling and Numerical Techniques'', June 18  22, 2018, Budapest, Hungary, June 18, 2018.

R. Lasarzik, Generalised solutions to the EricksenLeslie model describing liquid crystal flow, Università degli Studi di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, June 7, 2018.

R. Lasarzik, Generalized solution concepts to the EricksenLeslie equations modeling liquid crystal flow, 28th IFIP TC 7 Conference on System Modelling and Optimization, July 23  27, 2018, Universität DuisburgEssen, Essen, July 26, 2018.

M. Marschall, Bayesian inversion and adaptive lowrank tensor decomposition, 3rd GAMM AGUQ Workshop on Uncertainty Quantification 2018, TU Darmstadt, March 12  14, 2018.

M. Marschall, Bayesian inversion with adaptive lowrank approximation, Workshop of the FrenchGermanItalian LIA (Laboratoire International Associe) COPDESC on Applied Analysis ``Analysis, Control and Inverse Problems for PDEs'', November 26  30, 2018, University of Naples Federico II and Accademia Pontaniana, Italy, November 29, 2018.

M. Marschall, Bayesian inversion with adaptive lowrank approximation, Tensormeeting, November 26  27, 2018, Technische Universität Braunschweig, November 26, 2018.

M. Marschall, Baysian inversion and adaptive lowrank tensor decomposition, Workshop ``UQ for Inverse Problems in Complex Systems'', Cambridge, UK, April 9  13, 2018.

M. Marschall, Surrogate model adaption by explicit Bayesian inversion in tensor train format, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), March 19  23, 2018, Technische Universität München, March 21, 2018.
Forschungsgruppen
 Partielle Differentialgleichungen
 Laserdynamik
 Numerische Mathematik und Wissenschaftliches Rechnen
 Nichtlineare Optimierung und Inverse Probleme
 Stochastische Systeme mit Wechselwirkung
 Stochastische Algorithmen und Nichtparametrische Statistik
 Thermodynamische Modellierung und Analyse von Phasenübergängen
 Nichtglatte Variationsprobleme und Operatorgleichungen