• R. Henrion, Chapter 2: Calmness as a constraint qualification for M-stationarity 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. 21--41, (Chapter Published), DOI 10.1007/978-981-10-4774-9 .

  • L. Ghezzi, D. Hömberg, Ch. Landry, eds., Math for the Digital Factory, 27 of Mathematics in Industry / The European Consortium for Mathematics in Industry, Springer International Publishing AG, Cham, 2017, x+348 pages, (Collection Published), DOI 10.1007/978-3-319-63957-4 .

  • F.J. Aragón Artacho, R. Henrion, M.A. López-Cerdá, C. Sagastizábal, J.M. Borwein, eds., Special Issue: Advances in Monotone Operators Theory and Optimization, 25, issues 3 and 4, of Set-Valued and Variational Analysis, Springer International Publishing AG, Cham, 2017, 396 pages, (Collection Published).

Articles in Refereed Journals

  • M. Eigel, J. Neumann, R. Schneider, S. Wolf, Risk averse stochastic structural topology optimization, Computer Methods in Applied Mechanics and Engineering, 334 (2018), pp. 470--482, DOI 10.1016/j.cma.2018.02.003 .
    A novel approach for risk-averse 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 high-dimensional 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 risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional 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, Sampling-free 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. 035010/1--035010/29, DOI 10.1088/1361-6420/aaa998 .
    The statistical Bayesian approach is a natural setting to resolve the ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a sampling-free 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 sampling-free 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 high-dimensional quadrature of the log-likelihood. 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 low-rank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results.

  • M. Eigel, J. Neumann, R. Schneider, S. Wolf, Non-intrusive tensor reconstruction for high dimensional random PDEs, Computational Methods in Applied Mathematics, (2018), published online on 25.07.2018, DOI 10.1515/cmam-2018-0028 .
    This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank 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 black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling.

  • F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G.M. Schoenmakers, SDE based regression for random PDEs, SIAM Journal on Scientific Computing, 39 (2017), pp. A1168--A1200.
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.

  • F. Anker, Ch. Bayer, M. Eigel, J. Neumann, J.G.M. Schoenmakers, A fully adaptive interpolated stochastic sampling method for linear random PDEs, International Journal for Uncertainty Quantification, 7 (2017), pp. 189--205, DOI 10.1615/Int.J.UncertaintyQuantification.2017019428 .
    A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.

  • M. Eigel, R. Müller, A posteriori error control for stationary coupled bulk-surface equations, IMA Journal of Numerical Analysis, 38 (2018), pp. 271--298 (published online on 09.03.2017), DOI 10.1093/imanum/drw080 .
    We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm with several benchmark examples.

  • M. Eigel, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Numerische Mathematik, 136 (2017), pp. 765--803.
    The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated.

  • M. Eigel, K. Sturm, Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation, Optimization Methods & Software, 33 (2018), pp. 268--296 (published online on 03.05.2017), DOI 10.1080/10556788.2017.1314471 .
    In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.

  • T. González Grandón, H. Heitsch, R. Henrion, A joint model of probabilistic/robust constraints for gas transport management in stationary networks, Computational Management Science, 14 (2017), pp. 443--460, DOI 10.1007/s10287-017-0284-7 .
    We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.

  • F. Lanzara, V. Maz'ya, G. Schmidt, A fast solution method for time dependent multidimensional Schrödinger equations, Applicable Analysis. An International Journal, published online on 08.08.2017, url, DOI 10.1080/00036811.2017.1359571 .
    In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.

  • T. Abbas, H. Ammari, G. Hu, A. Wahab, J.Ch. Ye, Elastic scattering coefficients and enhancement of nearly elastic cloaking, Journal of Elasticity, 128 (2017), pp. 203--243, DOI 10.1007/s10659-017-9624-7 .
    The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic and electromagnetic media. In view of their promising applications, we introduce the notion of scattering coefficients of an elastic inclusion in this article. First, we define elastic scattering coefficients and substantiate that they naturally appear in the expansions of elastic scattered field and far field scattering amplitudes corresponding to a plane wave incidence. Then an algorithm is developed and analyzed for extracting the elastic scattering coefficients from multi-static response measurements of the scattered field. Moreover, the estimate of the maximal resolving order is provided in terms of the signal-to-noise ratio. The decay rate and symmetry of the elastic scattering coefficients are also discussed. Finally, we design scattering-coefficients-vanishing structures and elucidate their utility for enhancement of nearly elastic cloaking.

  • L. Adam, R. Henrion, J. Outrata, On M-stationarity conditions in MPECs and the associated qualification conditions, Mathematical Programming. A Publication of the Mathematical Programming Society, 168 (2018), pp. 229--259 (published online on 18.4.2017), DOI 10.1007/s10107-017-1146-3 .
    Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed constraint qualifications (CQs) as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible CQs, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive so-called M-stationarity conditions. The strength of assumptions and conclusions in the two forms of the MPEC is strongly related with the CQs on the 'lower level' imposed on the set whose normal cone appears in the generalized equation. For instance, under just the Mangasarian-Fromovitz CQ (a minimum assumption required for this set), the calmness properties of the original and the enhanced perturbation mapping are drastically different. They become identical in the case of a polyhedral set or when adding the Full Rank CQ. On the other hand, the resulting optimality conditions are affected too. If the considered set even satisfies the Linear Independence CQ, both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. A compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions is provided in the main Theorem 4.3. The obtained results are finally applied to MPECs with structured equilibria.

  • A.L. Diniz, R. Henrion, On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions, Energy Systems, 8 (2017), pp. 149--167, DOI 10.1007/s12667-015-0180-6 .

  • M.H. Farshbaf Shaker, R. Henrion, D. Hömberg, Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization, Set-Valued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., pp. published online on 11.10.2017, url, DOI 10.1007/s11228-017-0452-5 .
    Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints.

  • V. Guigues, R. Henrion, Joint dynamic probabilistic constraints with projected linear decision rules, Optimization Methods & Software, 32 (2017), pp. 1006--1032.
    We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.

  • T. Yin, A. Rathsfeld, L. Xu, A BIE-based DtN-FEM for fluid-solid interaction problems, Journal of Computational Mathematics, 36 (2018), pp. 47--69 (published online on 11.10.2017), DOI 10.4208/jcm.1610-m2015-0480 .

  • W. VAN Ackooij, R. Henrion, (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution, SIAM/ASA Journal on Uncertainty Quantification, 5 (2017), pp. 63--87, DOI 10.1137/16M1061308 .
    We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven.

  • D. Hömberg, F.S. Patacchini, K. Sakamoto, J. Zimmer, A revisited Johnson--Mehl--Avrami--Kolmogorov model and the evolution of grain-size distributions in steel, IMA Journal of Applied Mathematics, 82 (2017), pp. 763--780, DOI 10.1093/imamat/hxx012 .
    The classical Johnson-Mehl-Avrami-Kolmogorov approach for nucleation and growth models of diffusive phase transitions is revisited and applied to model the growth of ferrite in multiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the grain structure is essential. To this end, a Fokker-Planck evolution law for the volume distribution of ferrite grains is developed and shown to exhibit a log-normally distributed solution. Numerical parameter studies are given and confirm expected properties qualitatively. As a preparation for future work on parameter identification, a strategy is presented for the comparison of volume distributions with area distributions experimentally gained from polished micrograph sections.

Contributions to Collected Editions

  • 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. 831--843.

  • L. Capone, Th. Petzold, D. Hömberg, D. Ivanov, A novel integrated tool for parametric geometry generation and simulation for induction hardening of gears, in: Heat Treat 2017: Proceedings of the 29th AMS Heat Treating Society Conference, October 24--26, 2017, Columbus, Ohio, USA, ASM International, Materials Park, Ohio, 2017, pp. 234--242.

  • P. Das, J.I. Asperheim, B. Grande, Th. Petzold, D. Hömberg, Simulation of temperature profile in longitudinal welded tubes during high-frequency induction (HFI) welding, in: Heat Treat 2017: Proceedings of the 29th AMS Heat Treating Society Conference, October 24--26, 2017, Columbus, Ohio, USA, ASM International, Materials Park, Ohio, 2017, pp. 534--538.

Preprints, Reports, Technical Reports

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

  • M. Eigel, M. Marschall, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM for log-normal coefficients in hierarchical tensor representations, Preprint no. 2515, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2515 .
    Abstract, PDF (558 kByte)
    Stochastic Galerkin methods for non-affine 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 log-normal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted 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 residual-based 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 pipe-wise 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 Ericksen--Leslie system equipped with the Oseen--Frank energy, Preprint no. 2508, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2508 .
    Abstract, PDF (315 kByte)
    We analyze the Ericksen-Leslie 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 measure-valued solutions to the considered system and showed global existence as well as weak-strong 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 weak-strong 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 long-time behavior and show that all solutions converge for the large time limit to a certain steady state.

  • 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 broad-area laser, Preprint no. 2502, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2502 .
    Abstract, PDF (384 kByte)
    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 lower-order transversal optical modes in the laser diode and, consequently, improves the spatial beam quality.

  • R. Henrion, W. Römisch, Problem-based optimal scenario generation and reduction in stochastic programming, Preprint no. 2485, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2485 .
    Abstract, PDF (259 kByte)
    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 two-stage stochastic programs we show that the problem-based 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 semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

  • A. Hantoute, R. Henrion, P. Pérez-Aros, Subdifferential characterization of probability functions under Gaussian distribution, Preprint no. 2478, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2478 .
    Abstract, PDF (282 kByte)
    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 infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial 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.

  • R. Lasarzik, Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen--Frank 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 measure-valued solutions to the Ericksen-Leslie system equipped with the Oseen--Frank 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 measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.

  • R. Lasarzik, Weak-strong uniqueness for measure-valued solutions to the Ericksen--Leslie model equipped with the Oseen--Frank free energy, Preprint no. 2474, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2474 .
    Abstract, PDF (323 kByte)
    We analyze the Ericksen-Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i.e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong 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.

  • I. Franović, O.E. Omel'chenko, M. Wolfrum, Phase sensitive excitability of a limit cycle, Preprint no. 2465, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2465 .
    Abstract, PDF (5922 kByte)
    The classical notion of excitability refers to an equilibrium state that shows under the influence of perturbations a nonlinear threshold-like behavior. Here, we extend this concept by demonstrating how periodic orbits can exhibit a specific form of excitable behavior where the nonlinear threshold-like response appears only after perturbations applied within a certain part of the periodic orbit, i.e the excitability happens to be phase sensitive. As a paradigmatic example of this concept we employ the classical FitzHugh-Nagumo system. The relaxation oscillations, appearing in the oscillatory regime of this system, turn out to exhibit a phase sensitive nonlinear threshold-like response to perturbations, which can be explained by the nonlinear behavior in the vicinity of the canard trajectory. Triggering the phase sensitive excitability of the relaxation oscillations by noise we find a characteristic non-monotone dependence of the mean spiking rate of the relaxation oscillation on the noise level. We explain this non-monotone dependence as a result of an interplay of two competing effects of the increasing noise: the growing efficiency of the excitation and the degradation of the nonlinear response.

Talks, Poster

  • M.J. Arenas Jaén, Modelling, simulation and optimization of inductive pre-heating 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, Cambridge, UK, February 7, 2018.

  • R. Gruhlke, Domain decomposition and multi-scale 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, Stochastic domain decomposition and application in heterogeneous material, 3rd GAMM AGUQ Workshop on Uncertainty Quantification 2018, TU Dortmund, March 12 - 14, 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 SQP-solver 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 high-dimensional 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), Minisymposia 96, 109, 122 ``Low-Rank Approximations for the Forward- and the Inverse Problems I--III'', April 16 - 19, 2018, Garden Grove, USA, April 19, 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, M-stationary 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 with probust constraints, Colloquium & International Conference on Variational Analysis and Nonsmooth Optimization, June 28 - July 1, 2018, Martin-Luther-Universität Halle-Wittenberg, Halle, June 28, 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, 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, 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, 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 Ericksen-Leslie model describing liquid crystal flow, Università degli Studi di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, June 7, 2018.

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

  • M. Marschall, Baysian inversion and adaptive low-rank 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.

  • M.J. Arenas Jaén, Modelling, simulation and optimization of inductive pre- and post-heating for thermal cutting of steel plates, Midterm-Meeting MIMESIS -- Mathematics and Materials Science for Steel Production and Manufacturing, WIAS Berlin, September 18, 2017.

  • L. Capone, Simulation and optimization of induction hardening for bevel and helical gears, Midterm-Meeting MIMESIS -- Mathematics and Materials Science for Steel Production and Manufacturing, WIAS Berlin, September 18, 2017.

  • M. Eigel, A sampling-free adaptive Bayesian inversion with hierarchical tensor representations, European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2017), Minisymposium 15 ``Uncertainty Propagation'', September 25 - 29, 2017, Voss, Norway, September 27, 2017.

  • M. Eigel, Adaptive stochastic FE for explicit Bayesian inversion with hierarchical tensor representations, Institut National de Recherche en Informatique et en Automatique (INRIA), SERENA (Simulation for the Environment: Reliable and Efficient Numerical Algorithms) research team, Paris, France, June 1, 2017.

  • M. Eigel, Adaptive stochastic Galerkin FE and tensor compression for random PDEs, sc Matheon Workshop ``Reliable Methods of Mathematical Modeling'' (RMMM8), July 31 - August 4, 2017, Humboldt-Universität zu Berlin, August 3, 2017.

  • M. Eigel, Aspects of stochastic Galerkin FEM, Universität Basel, Mathematisches Institut, Switzerland, November 10, 2017.

  • M. Eigel, Efficient Bayesian inversion with hierarchical tensor representation, 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2017), June 15 - 17, 2017, Rhodos, Greece, June 16, 2017.

  • M. Eigel, Explicit Bayesian inversion in hierarchical tensor representations, 4th GAMM Junior's and 1st GRK2075 Summer School 2017 ``Bayesian Inference: Probabilistic Way of Learning from Data'', July 10 - 14, 2017, Braunschweig, July 14, 2017.

  • M. Eigel, Stochastic topology optimization with hierarchical tensor reconstruction, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6 - 8, 2017, München, September 7, 2017.

  • R. Gruhlke, Annual Report 2017 MuScaBlaDes (Subproject 4 ``Multi-scale failure analysis with polymorphic uncertainties for optimal design of rotor blades''), Jahrestreffen des SPP 1886, October 12 - 13, 2017, Technische Universität München, October 13, 2017.

  • R. Gruhlke, Multi-scale failure analysis with polymorphic uncertainties for optimal design of rotor blades, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6 - 8, 2017, München, September 6, 2017.

  • H. Heitsch, A probabilistic approach to optimization problems in gas transport networks, SESO 2017 International Thematic Week ``Smart Energy and Stochastic Optimization'', May 30 - June 1, 2017, ENSTA ParisTech and École des Ponts ParisTech, Paris, France, June 1, 2017.

  • H. Heitsch, A probabilistic approach to optimization problems in gas transport networks, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 6, 2017.

  • H. Heitsch, On probabilistic capacity maximization in stationary gas networks, 21st Conference of the International Federation of Operational Research Societies (IFORS 2017), Invited Session TB20 ``Optimization of Gas Networks 2'', July 17 - 21, 2017, Quebec, Canada, July 18, 2017.

  • J. Neumann, Topology optimization under uncertainties, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S15 ``Uncertainty Quantification'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

  • R. Henrion, A friendly tour through the world of calmness, 11th International Conference on Parametric Optimization and Related Topics (ParaoptXI), September 19 - 22, 2017, Prague, Czech Republic, September 19, 2017.

  • R. Henrion, Comparing and verifying calmness conditions for MPECs, Second Workshop on Metric Bounds and Transversality (WoMBaT 2017), November 30 - December 2, 2017, RMIT University, Melbourne, Australia, November 30, 2017.

  • R. Henrion, Contraintes en probabilité: Formules du gradient et applications, Workshop ``MAS-MODE 2017'', Institut Henri Poincaré, Paris, France, January 9, 2017.

  • R. Henrion, On M-stationary condition for a simple electricity spot market model, Workshop ``Variational Analysis and Applications for Modelling of Energy Exchange'', May 4 - 5, 2017, Université Perpignan, France, May 4, 2017.

  • R. Henrion, On a joint model for probabilistic/robust constraints with an application to gas networks under uncertainties, Workshop ``Models and Methods of Robust Optimization'', March 9 - 10, 2017, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM, Kaiserslautern, March 10, 2017.

  • R. Henrion, Optimization problems under robust constraints with applications to gas networks under uncertainty, The Eighth Australia-China Workshop on Optimization (ACWO 2017), December 4, 2017, Curtin University, Perth, Australia, December 4, 2017.

  • R. Henrion, Probabilistic constraints in infinite dimensions, Universität Wien, Institut für Statistik und Operations Research, Austria, November 6, 2017.

  • R. Henrion, Probabilistic constraints: Convexity issues and beyond, XII International Symposium on Generalized Convexity and Monotonicity, August 27 - September 2, 2017, Hajdúszoboszló, Hungary, August 29, 2017.

  • R. Henrion, Probabilistic programming in infinite dimensions, The South Pacific Optimization Meeting in Western Australia 2017 (SPOM in WA 2017), December 8 - 10, 2017, Curtin University, Perth, Australia, December 9, 2017.

  • R. Henrion, Probabilistic programming: Structural properties and applications, Control and Optimization Conference on the occasion of Frédéric Bonnans 60th birthday, November 15 - 17, 2017, Electricité de France, Palaiseau, France, November 17, 2017.

  • R. Henrion, Problèmes d'optimisation sous contraintes en probabilité, Université de Bourgogne, Département de Mathématiques, Dijon, France, October 25, 2017.

  • R. Henrion, Subdifferential characterization of Gaussian probability functions, SESO 2017 International Thematic Week ``Smart Energy and Stochastic Optimization'', May 30 - June 1, 2017, ENSTA ParisTech and École des Ponts ParisTech, Paris, France, June 1, 2017.

  • R. Henrion, Subdifferential estimates for Gaussian probability functions, HCM Workshop: Nonsmooth Optimization and its Applications, May 15 - 19, 2017, Hausdorff Center for Mathematics, Bonn, May 17, 2017.

  • R. Henrion, Subdifferential of probability functions under Gaussian distribution, The Second Pacific Optimization Conference (POC2017), December 4 - 7, 2017, Curtin University, Perth, Australia, December 6, 2017.

  • D. Hömberg, European collaboration in industrial and applied mathematics, 25th Conference on Applied and Industrial Mathematics (CAIM), September 14 - 17, 2017, University of Iaşi, Romania, September 14, 2017.

  • D. Hömberg, From dilatometer experiments to distortion compensation -- Optimal control problems related to solid-solid phase transitions, Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway, November 3, 2017.

  • D. Hömberg, Joule heating models -- Modelling, analysis and industrial application, Beijing Computational Science Research Center, China, October 10, 2017.

  • D. Hömberg, MIMESIS Mid-term Meeting -- Coordinator's report, Midterm-Meeting MIMESIS -- Mathematics and Materials Science for Steel Production and Manufacturing, WIAS Berlin, September 18, 2017.

  • D. Hömberg, MSO for steel production and manufacturing, Workshop ``Future and Emerging Mathematical Technologies in Europe'', December 11 - 15, 2017, Lorentz Center, Leiden, Netherlands, December 11, 2017.

  • D. Hömberg, Mathematical aspects of multi-frequency induction heating, Universidade Técnica de Lisboa, Instituto Superior Técnico, Portugal, February 2, 2017.

  • D. Hömberg, On a robust phase field approach to topology optimization, Università degli Studi di Pavia, Dipartimento di Matematica, Italy, April 28, 2017.

  • D. Hömberg, Optimal coefficient control for semilinear parabolic equations, The 15th Annual Meeting of the China Society for Industrial and Applied Mathematics, Embedded Meeting EM02 ``A3 Workshop on Modeling and Computation of Applied Inverse Problems'', October 12 - 15, 2017, Qingdao, China, October 14, 2017.

  • D. Hömberg, The Digital Factory -- A perspective for a closer cooperation between Math and Industry, Meeting ``M414 Mathematics for Industry 4.0'', November 7, 2017, Vicenza Convention Centre, Italy, November 7, 2017.

  • M. Marschall, Bayesian inversion using hierarchical tensors, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S15 ``Uncertainty Quantification'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 8, 2017.

  • M. Marschall, Sampling-free Bayesian inversion with adaptive hierarchical tensor representation, Frontiers of Uncertainty Quantification in Engineering (FrontUQ 2017), September 6 - 8, 2017, München, September 7, 2017.

  • M. Marschall, Sampling-free Bayesian inversion with adaptive hierarchical tensor representation, International Conference on Scientific Computation and Differential Equations (SciCADE2017), MS21 ``Tensor Approximations of Multi-Dimensional PDEs'', September 11 - 15, 2017, University of Bath, UK, September 14, 2017.

  • A. Rathsfeld, Neue Software und Simulationsergebnisse, LayTec AG, Berlin, June 9, 2017.