Articles in Refereed Journals

  • 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.
    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.

  • M. Eigel, Ch. Merdon, J. Neumann, An adaptive multilevel Monte--Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016) pp. 1219--1245.
    The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.

  • M. Eigel, Ch. Merdon, Equilibration a posteriori error estimation for convection-diffusion-reaction problems, Journal of Scientific Computing, 67 (2016) pp. 747--768.
    We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved.

    Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases.

  • M. Eigel, Ch. Merdon, Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin finite element methods, SIAM/ASA Journal on Uncertainty Quantification, 4 (2016) pp. 1372--1397.
    Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process.

  • H. Heitsch, H. Leövey, W. Römisch, Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?, Computational Optimization and Applications. An International Journal, 65 (2016) pp. 567--603.
    Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.

  • G. Hu, M. Salo, E.V. Vesalainen, Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM Journal on Mathematical Analysis, 48 (2016) pp. 152--165.

  • G. Hu, A. Kirsch, T. Yin, Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves, Inverse Problems and Imaging, 10 (2016) pp. 103--129.
    Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with homogeneous compressible inviscid fluid with a constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by the Lamé constants. This paper is concerned with direct (or forward) and inverse fluid-solid interaction (FSI) problems with unbounded bi-periodic interfaces between acoustic and elastic waves. We present a variational approach to the forward interaction problem with Lipschitz interfaces. Existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Concerning the inverse problem, we show that the factorization method by Kirsch (1998) is applicable to the FSI problem in periodic structures. A computational criterion and a uniqueness result are justified for precisely characterizing the elastic body by utilizing the scattered acoustic near field measured in the fluid.

  • M.J. Cánovas, R. Henrion, M.A. López, J. Parra, Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming, Journal of Optimization Theory and Applications, 169 (2016) pp. 925--952.
    With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.

  • M.J. Cánovas, R. Henrion, J. Parra, F.J. Toledo, Critical objective size and calmness modulus in linear programming, Set-Valued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., 24 (2016) pp. 565--579.
    This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in citeCHPTmp. In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds.

  • C. Gotzes, H. Heitsch, R. Henrion, R. Schultz, Feasibility of nominations in stationary gas networks with random load, Mathematical Methods of Operations Research, 84 (2016) pp. 427--457.
    The paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size.

  • F. Lanzara, V. Maz'ya, G. Schmidt, Approximation of solutions to multidimensional parabolic equations by approximate approximations, Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, 41 (2016) pp. 749--767.

  • C. Carstensen, M. Eigel, Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes, Computational Methods in Applied Mathematics, 16 (2016) pp. 213--230.
    A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to pi/2. The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

  • G. Colombo, R. Henrion, N.D. Hoang, B.S. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, Journal of Differential Equations, 260 (2016) pp. 3397--3447.
    The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W1,2-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.

  • Y. Guo, D. Hömberg, G. Hu, J. Li, H. Liu, A time domain sampling method for inverse acoustic scattering problems, Journal of Computational Physics, 314 (2016) pp. 647--660.
    This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally ``direct'' involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.

  • G. Hu, A. Rathsfeld, T. Yin, Finite element method to fluid-solid interaction problems with unbounded periodic interfaces, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016) pp. 5--35.
    Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

  • K. Sturm, M. Hintermüller, D. Hömberg, Distortion compensation as a shape optimisation problem for a sharp interface model, Computational Optimization and Applications. An International Journal, 64 (2016) pp. 557--588.
    We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.

  • T. Yin, G. Hu, L. Xu, B. Zhang, Near-field imaging of obstacles with the factorization method: Fluid-solid interaction, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 32 (2016) pp. 015003/1--015003/29.

  • M.H. Farshbaf Shaker, C. Hecht, Optimal control of elastic vector-valued Allen--Cahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016) pp. 129--152.
    In this paper we consider a elastic vector-valued Allen--Cahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized first-order necessary optimality conditions for the original problem.

  • D. Hömberg, Q. Liu, J. Montalvo-Urquizo, D. Nadolski, Th. Petzold, A. Schmidt, A. Schulz, Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects, Finite Elements in Analysis and Design, 121 (2016) pp. 86--100.
    Induction hardening is a well known method for the heat treatment of steel components. With the concept of multi-frequency hardening, where currents with two different frequency components are provided on a single inductor coil, it is possible to optimize the hardening zone to follow a given contour, e.g. of a gear. In this article, we consider the simulation of multi-frequency induction hardening in 3D. The equations to solve are the vector potential formulation of Maxwell's equations describing the electromagnetic fields, the balance of momentum to determine internal stresses and deformations arising from thermoelasticity and transformation induced plasticity, a rate law to determine the distribution of different phases and the heat equation to determine the temperature distribution in the workpiece. The equations are solved using adaptive finite element methods. The simulation results are compared to experiments for discs and for gears. A very good agreement for the hardening profile and the temperature is observed. It is also possible to predict the distribution of residual stresses after the heat treatment.

Contributions to Collected Editions

  • Q. Liu, Th. Petzold, D. Nadolski, R. Pulch, Simulation of thermomechanical behavior subjected to induction hardening, in: Scientific Computing in Electrical Engineering, SCEE 2014, Wuppertal, Germany, July 2014, A. Bartel, M. Clemens, M. Günther, E.J.W. TER Maten, eds., 23 of Mathematics in Industry, Springer International Publishing Switzerland, Cham, 2016, pp. 133--142.

Preprints, Reports, Technical Reports

  • D. Hömberg, O. Rott, K. Sturm, Discretisation and error analysis for a mathematical model of milling processes, Preprint no. 2364, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2364 .
    Abstract, PDF (565 kByte)
    We investigate a mathematical model for milling where the cutting tool dynamics is considered together with an elastic workpiece model. Both are coupled by the cutting forces consisting of two dynamic components representing vibrations of the tool and of the workpiece, respectively, at the present and previous tooth periods. We develop a numerical solution algorithm and derive error estimates both for the semi-discrete and the fully discrete numerical scheme. Numerical computations in the last section support the analytically derived error estimates.

  • M. Eigel, M. Marschall, R. Schneider, Bayesian inversion with a hierarchical tensor representation, Preprint no. 2363, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2363 .
    Abstract, PDF (812 kByte)
    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, Stochastic topology optimisation with hierarchical tensor reconstruction, Preprint no. 2362, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2362 .
    Abstract, PDF (8552 kByte)
    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.

  • G. Schmidt, Scattering of general incident beams by diffraction gratings, Preprint no. 2355, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2355 .
    Abstract, PDF (1036 kByte)
    The paper is devoted to the electromagnetic scattering of arbitrary time-harmonic fields by periodic structures. The Floquet-Fourier transform converts the full space Maxwell problem to a two-parameter family of diffraction problems with quasiperiodic incidence waves, for which conventional grating methods become applicable. The inverse transform is given by integrating with respect to the parameters over a infinite strip in R^2. For the computation of the scattered fields we propose an algorithm, which extends known adaptive methods for the approximate calculation of multiple integrals. The novel adaptive approach provides autonomously the expansion of the incident field into quasiperiodic waves in order to approximate the scattered fields within a prescribed error tolerance. Some application examples are numerically examined.

  • J. Elschner, G. Hu, M. Yamamoto, Single logarithmic conditional stability in determining unknown boundaries, Preprint no. 2351, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2351 .
    Abstract, PDF (442 kByte)
    We prove a conditional stability estimate of log-type for determining unknown boundaries from a single Cauchy data taken on an accessible subboundary. Our approach relies on new interior and boundary estimates derived from the Carleman estimate for elliptic equations. A local stability result for target identification of an acoustic sound-soft scatterer from a single far-field pattern is also obtained.

  • G. Bao, G. Hu, T. Yin, J. Sun, Direct and inverse elastic scattering from anisotropic media, Preprint no. 2348, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2348 .
    Abstract, PDF (1194 kByte)
    Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the scattering interface is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are demonstrated to show the validity and accuracy of our methods.

  • M.H. Farshbaf Shaker, R. Henrion, D. Hömberg, Chance constraints in PDE constrained optimization, Preprint no. 2338, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2338 .
    Abstract, PDF (217 kByte)
    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.

  • 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, Preprint no. 2283, WIAS, Berlin, 2016.
    Abstract, PDF (4023 kByte)
    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.

  • F. Lanzara, V. Maz'ya, G. Schmidt, A fast solution method for time dependent multidimensional Schrödinger equations, Preprint no. 2279, WIAS, Berlin, 2016.
    Abstract, PDF (3794 kByte)
    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, Preprint no. 2275, WIAS, Berlin, 2016.
    Abstract, PDF (311 kByte)
    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.

  • V. Guigues, R. Henrion, Joint dynamic probabilistic constraints with projected linear decision rules, Preprint no. 2271, WIAS, Berlin, 2016.
    Abstract, PDF (413 kByte)
    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.

  • M.H. Farshbaf Shaker, Ch. Heinemann, Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D, Preprint no. 2269, WIAS, Berlin, 2016.
    Abstract, PDF (247 kByte)
    Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749--2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.

  • M. Eigel, K. Sturm, Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation, Preprint no. 2244, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2244 .
    Abstract, PDF (6274 kByte)
    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.

  • J. Elschner, G. Hu, Acoustic scattering from corners, edges and circular cones, Preprint no. 2242, WIAS, Berlin, 2016.
    Abstract, PDF (397 kByte)
    Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be non-convex. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone.

  • L. Adam, R. Henrion, J. Outrata, On M-stationarity conditions in MPECs and the associated qualification conditions, Preprint no. 2215, WIAS, Berlin, 2016.
    Abstract, PDF (257 kByte)
    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.

Talks, Poster

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

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

  • N. Togobytska, Uncertainty quantification in environmental simulations, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 26 ``Stochastic Inverse Problems'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 13, 2016.

  • M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016), January 5 - 10, 2016, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, January 8, 2016.

  • M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, 15th Conference on the Mathematics of Finite Elements and Applications (Brunel MAFELAP 2016), Minisymposium ``Uncertainty Quantification Using Stochastic PDEs and Finite Elements'', June 14 - 17, 2016, Brunel University London, Uxbridge, UK, June 14, 2016.

  • M. Eigel, Adaptive stochastic Galerkin FEM with hierarchical tensor representations, Joint Annual Meeting of DMV and GAMM, Section 18 ``Numerical Methods of Differential Equations'', March 7 - 11, 2016, Technische Universität Braunschweig, March 10, 2016.

  • M. Eigel, Bayesian inversion using hierarchical tensor approximations, SIAM Conference on Uncertainty Quantification, Minisymposium 67 ``Bayesian Inversion and Low-rank Approximation (Part II)'', April 5 - 8, 2016, Lausanne, Switzerland, April 6, 2016.

  • M. Eigel, Some aspects of adaptive random PDEs, Oberseminar, Rheinisch-Westfälische Technische Hochschule Aachen, Institut für Geometrie und Praktische Mathematik, July 21, 2016.

  • H. Heitsch, Nonlinear probabilistic constraints in gas transportation problems, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, Australia, May 11, 2016.

  • H. Heitsch, Optimization in gas transport networks using nonlinear probabilistic constraints, XIV International Conference on Stochastic Programming (ICSP 2016), Thematic Session: Probabilistic Constraints: Applications and Theory, June 25 - July 1, 2016, Búzios, Brazil, June 28, 2016.

  • G. Hu, Direct and inverse problems in elastodynamics, Workshop ``Theory, Numerics and Application of Partial Differential Equations'', December 10, 2016, Chinese Academy of Sciences, Beijing, December 10, 2016.

  • G. Hu, Inverse medium scattering problems, Workshop on Inverse Problems and their Applications, November 18 - 20, 2016, Southeast University, Nanjing, China, November 20, 2016.

  • G. Hu, Uniqueness in inverse medium scattering problems, Workshop ``Theory and Numerics of Inverse Scattering Problems'', September 18 - 24, 2016, Mathematisches Forschungsinstitut Oberwolfach, September 20, 2016.

  • J. Neumann, Adaptive SDE based sampling for random PDE, SIAM Conference on Uncertainty Quantification, Minisymposium 142 ``Error Estimation and Adaptive Methods for Uncertainty Quantification in Computational Sciences -- Part II'', April 5 - 8, 2016, Lausanne, Switzerland, April 8, 2016.

  • J. Neumann, The phase field approach for topology optimization under uncertainties, ZIB Computational Medicine and Numerical Mathematics Seminar, Konrad-Zuse-Zentrum für Informationstechnik Berlin, August 25, 2016.

  • TH. Petzold, The MIMESIS project -- An example for an interdisciplinary research project, Leibniz-Kolleg for Young Researchers: Chances and Challenges of Interdisciplinary Research, Thematic Workshop ``Models and Modelling'', November 9 - 11, 2016, Leibniz-Gemeinschaft, Berlin, November 9, 2016.

  • I. Bremer, Dealing with probabilistic constraints under multivariate normal distribution in a standard SQP solver by using Genz' method, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • M.H. Farshbaf Shaker, A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20 - 24, 2016, Cortona, Italy, June 21, 2016.

  • M.H. Farshbaf Shaker, Allen--Cahn MPECs, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • R. Henrion, (Sub-)Gradient formulae for Gaussian probability functions, XIV International Conference on Stochastic Programming (ICSP 2016), Thematic Session: Probabilistic Constraints: Applications and Theory, June 25 - July 1, 2016, Búzios, Brazil, June 28, 2016.

  • R. Henrion, Aspects of nondifferentiability for probability functions, 7th International Seminar on Optimization and Variational Analysis, June 1 - 3, 2016, Universidad de Alicante, Spain, June 2, 2016.

  • R. Henrion, Aspects of nonsmoothness for Gaussian probability functions, PGMO Days 2016 -- Gaspard Monge Program for Optimization and Operations Research, November 8 - 9, 2016, Electricité de France, Palaiseau, France, November 9, 2016.

  • R. Henrion, Calmness of the perturbation mappings for MPECs in original and enhanced form, International Conference on Bilevel Optimization and Related Topics, May 4 - 6, 2016, Dresden, May 6, 2016.

  • R. Henrion, Formules du gradient pour des fonctions probabilistes Gaussiennes, Workshop on Offshore Wind Generation, September 9, 2016, Electricité de France R&D, Paris, France, September 9, 2016.

  • R. Henrion, Initiation aux problèmes d'optimisation sous contraintes en probabilité, Workshop ``Optimisation en Milieu Aléatoire'', November 8, 2016, Institut des Sciences Informatiques et de leurs Interactions, GdR 720 ISIS (Information, Signal, Image et ViSion), Paristech Télécom, Paris, France, November 8, 2016.

  • R. Henrion, Optimisation sous contraintes en probabilité, Séminaire du Groupe de Travail ``Modèles Stochastiques en Finance'', Ecole Nationale Supérieure des Techniques Avancées (ENSTA) ParisTech, Palaiseau, France, November 28, 2016.

  • R. Henrion, Robust-stochastic optimization problems in stationary gas networks, Conference ``Mathematics of Gas Transport'', October 6 - 7, 2016, Zuse Institut Berlin, October 6, 2016.

  • D. Hömberg, Analysis and simulation of Joule heating problems, Mathematisches Kolloquium, Bergische Universität Wuppertal, Fachgruppe Mathematik und Informatik, June 21, 2016.

  • D. Hömberg, European Industrial Doctorates -- A funding opportunity for collaboration with industry, Math Meets Industry, September 22 - 23, 2016, Trondheim, Norway, September 22, 2016.

  • D. Hömberg, European collaboration in Industrial and Applied Mathematics, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 38 ``Maths in HORIZON 2020 and Beyond'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 15, 2016.

  • D. Hömberg, Math for steel production and manufacturing, MACSI10 -- Empowering Industrial Mathematical and Statistical Modelling for the Future, December 8 - 9, 2016, University of Limerick, Ireland, December 9, 2016.

  • D. Hömberg, Modelling and simulation of multi-frequency induction hardening, Fifth Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2016), January 11 - 15, 2016, Universidad de Concepción, Chile, January 14, 2016.

  • D. Hömberg, Multifrequency induction hardening -- Modelling and simulation, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Kaiserslautern, May 17, 2016.

  • G. Schmidt, Scattering of general incident beams by diffraction gratings, European Optical Society Annual Meeting (EOSAM) 2016, September 26 - 30, 2016, Berlin, September 28, 2016.