Nonlinear Optimization and Inverse Problems

Publications

Articles in Refereed Journals

• M.J. Arenas Jaén, D. Hömberg, R. Lasarzik, P. Mikkonen, Th. Petzold, Modelling and simulation of flame cutting for steel plates with solid phases and melting, Journal of Mathematics in Industry, 10 (2020), pp. 18/1--18/16, DOI 10.1186/s13362-020-00086-0 .
  Abstract
  The goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiebaud [1] and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed.

• M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, SIAM Journal on Control and Optimization, 58 (2020), pp. 2288--2311, DOI 10.1137/19M1269944 .
  Abstract
  In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

• M.J. Cánovas, M.J. Gisbert, R. Henrion, J. Parra, Lipschitz lower semicontinuity moduli for linear inequality systems, Journal of Mathematical Analysis and Applications, 2 (2020), pp. 124313/1--124313/21, DOI 10.1016/j.jmaa.2020.124313 .
  Abstract
  The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings.

• D. Adelhütte, D. Assmann, T. González Grandón, M. Gugat, H. Heitsch, R. Henrion, F. Liers, S. Nitsche, R. Schultz, M. Stingl, D. Wintergerst, Joint model of probabilistic-robust (probust) constraints with application to gas network optimization, Vietnam Journal of Mathematics, published online on 10.11.2020.
  Abstract
  Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.

• J.I. Asperheim, P. Das, B. Grande, D. Hömberg, Th. Petzold, Three-dimensional numerical study of heat affected zone in induction welding of tubes, COMPEL. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. Emerald, Bradford, West Yorkshire. English, English abstracts., 39 (2020), pp. 213--219, DOI 10.1108/COMPEL-06-2019-0238 .

• M. Eigel, M. Marschall, M. Multerer, An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains, SIAM/ASA Journal on Uncertainty Quantification, 8 (2020), pp. 1189--1214 (published online on 25.08.2020), DOI 10.1137/19M1246080 .
  Abstract
  A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Loeve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems.

• M. Eigel, M. Marschall, M. Pfeffer, R. Schneider, Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations, Numerische Mathematik, 145 (2020), pp. 655--692, DOI 10.1007/s00211-020-01123-1 .
  Abstract
  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, On probabilistic capacity maximization in a stationary gas network, Optimization. A Journal of Mathematical Programming and Operations Research, 69 (2020), pp. 575--604 (published online on 10.06.2019), DOI 10.1080/02331934.2019.1625353 .
  Abstract
  The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach to assist gas network operators in managing uncertainty with respect to the demand and in exposing free network capacities while increasing reliability of transmission and supply. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving an optimization problem with joint probabilistic constraints over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing free booked capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. We are going to validate a kind of rank two constraint qualification implying the differentiability of the considered capacity problem. At the numerical side we are going to solve the problem using a projected gradient decent method, where the function and gradient evaluations of the probabilistic constraints are performed by the approach of spheric-radial decomposition applied for multivariate Gaussian random variables and more general distributions.

• M. Carraturo, E. Rocca, E. Bonetti, D. Hömberg, A. Reali, F. Auricchio, Graded-material design based on phase-field and topology optimization, Computational Mechanics, 64 (2019), pp. 1589--1600, DOI 10.1007/s00466-019-01736-w .
  Abstract
  In the present work we introduce a novel graded-material design for additive manufacturing based on phase-field and topology optimization. The main novelty of this work comes from the introduction of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Two different numerical examples are discussed, in both of them we perform sensitivity studies to asses the effects of different model parameters onto the resulting structure. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.

• E. Emmrich, R. Lasarzik, Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations, IMA Journal of Applied Mathematics, 84 (2019), pp. 1143--1176, DOI 10.1093/imamat/hxz030 .
  Abstract
  A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.

• I. Papaioannou, M. Daub, M. Drieschner, F. Duddeck, M. Ehre, L. Eichner, M. Eigel, M. Götz, W. Graf, L. Grasedyck, R. Gruhlke, D. Hömberg, M. Kaliske, D. Moser, Y. Petryna, D. Straub, Assessment and design of an engineering structure with polymorphic uncertainty quantification, GAMM-Mitteilungen, 42 (2019), pp. e201900009/1--e201900009/22, DOI 10.1002/gamm.201900009 .

• D. Pivovarov, K. Willner, P. Steinmann, S. Brumme, M. Müller, T. Srisupattarawanit, G.-P. Ostermeyer, C. Henning, T. Ricken, S. Kastian, S. Reese, D. Moser, L. Grasedyck, J. Biehler, M. Pfaller, W. Wall, Th. Kolsche, O. VON Estorff, R. Gruhlke, M. Eigel, M. Ehre, I. Papaioannou, D. Straub, S. Leyendecker, Challenges of order reduction techniques for problems involving polymorphic uncertainty, GAMM-Mitteilungen, 42 (2019), pp. e201900011/1--e201900011/24.

• W. VAN Ackooij, R. Henrion, P. Pérez-Aros, Generalized gradients for probabilistic/robust (probust) constraints, Optimization. A Journal of Mathematical Programming and Operations Research, 69 (2020), pp. 1451--1479 (published online on 14.02.2019), DOI 10.1080/02331934.2019.1576670 .
  Abstract
  Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data.

• M. Eigel, R. Schneider, P. Trunschke, S. Wolf, Variational Monte Carlo---Bridging concepts of machine learning and high dimensional partial differential equations, Advances in Computational Mathematics, 45 (2019), pp. 2503--2532, DOI 10.1007/s10444-019-09723-8 .
  Abstract
  A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.

• D. Hömberg, K. Krumbiegel, N. Togobytska, Optimal control of multiphase steel production, Journal of Mathematics in Industry, 9 (2019), pp. 1--32, DOI 10.1186/s13362-019-0063-x .
  Abstract
  An optimal control problem for the production of multiphase steel is investigated, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the ferrite phase fraction. The optimal control problem is analyzed and the first-order necessary and second-order sufficient optimality conditions are derived. For the numerical solution of the control problem reduced sequential quadratic programming (rSQP) method with a primal-dual active set strategy (PDAS) was applied. The numerical results were presented for the optimal control of a cooling line for production of hot rolled Mo-Mn dual phase steel.

• D. Hömberg, S. Lu, M. Yamamoto, Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data, Journal of Differential Equations, 266 (2019), pp. 7525--7544, DOI 10.1016/j.jde.2018.12.004 .
  Abstract
  We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.

• R. Lasarzik, Approximation and optimal control of dissipative solutions to the Ericksen--Leslie system, Numerical Functional Analysis and Optimization. An International Journal, 40 (2019), pp. 1721--1767, DOI 10.1080/01630563.2019.1632895 .
  Abstract
  We analyze the Ericksen--Leslie system equipped with the Oseen--Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relative simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, show the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.

• R. Lasarzik, Measure-valued solutions to the Ericksen--Leslie model equipped with the Oseen--Frank energy, Nonlinear Analysis. An International Mathematical Journal, 179 (2019), pp. 146--183, DOI 10.1016/j.na.2018.08.013 .
  Abstract
  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, Journal of Mathematical Analysis and Applications, 470 (2019), pp. 36--90, DOI 10.1016/j.jmaa.2018.09.051 .
  Abstract
  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.

Contributions to Collected Editions

• C. Brée, V. Raab, D. Gailevičius, V. Purlys, J. Montiel, G.G. Werner, K. Staliunas, A. Rathsfeld, U. Bandelow, M. Radziunas, Genetically optimized photonic crystal for spatial filtering of reinjection into broad-area diode lasers, in: 2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, OSA Technical Digest, IEEE, Piscataway, 2019, pp. 1--1, DOI 10.1109/CLEO-EQEC.2019.8871622 .
  Abstract
  Modern high-power broad-area semiconductor laser diodes (BASLDs) deliver optical output powers of several ten Watts at high electro-optical conversion efficiencies, which makes them highly relevant for numerous industrial, medical and scientific applications. However, lateral multimode behavior in BASLDs due to thermal lensing turns out highly detrimental, as it results in poor focusability and decreased laser beam brightnesss. Approaches to overcome this issue include improved epitaxial layer design, the optimization of evanescent spatial filtering by tailoring the emitter geometry and facet reflectivity, or Fourier spatially filtered reinjection from an external resonator [1].

• H. Heitsch, N. Strogies, Consequences of uncertain friction for the transport of natural gas through passive networks of pipelines, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 211--238.

Preprints, Reports, Technical Reports

• T. González Grandón, R. Henrion, P. Pérez-Aros, Dynamic probabilistic constraints under continuous random distributions, Preprint no. 2783, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2783 .
  Abstract, PDF (337 kByte)
  The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L ² is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

• M. Eigel, O. Ernst, B. Sprungk, L. Tamellini, On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion, Preprint no. 2753, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2753 .
  Abstract, PDF (325 kByte)
  Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting.

• H. Heitsch, R. Henrion, An enumerative formula for the spherical cap discrepancy, Preprint no. 2744, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2744 .
  Abstract, PDF (271 kByte)
  The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.

• R. Lasarzik, Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model, Preprint no. 2739, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2739 .
  Abstract, PDF (305 kByte)
  In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.

• G. Hu, A. Rathsfeld, Radiation conditions for the Helmholtz equation in a half plane filled by inhomogeneous periodic material, Preprint no. 2726, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2726 .
  Abstract, PDF (439 kByte)
  In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method.

• L. Baňas, R. Lasarzik, A. Prohl, Numerical analysis for nematic electrolytes, Preprint no. 2717, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2717 .
  Abstract, PDF (8587 kByte)
  We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

• M. Eigel, P. Trunschke, R. Schneider, Convergence bounds for empirical nonlinear least-squares, Preprint no. 2714, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2714 .
  Abstract, PDF (4747 kByte)
  We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.

• M. Eigel, R. Gruhlke, M. Marschall, Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion, Preprint no. 2672, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2672 .
  Abstract, PDF (769 kByte)
  A novel method for the accurate functional approximation of possibly highly concentrated probability densities is developed. It is based on the combination of several modern techniques such as transport maps and nonintrusive reconstructions of low-rank tensor representations. The central idea is to carry out computations for statistical quantities of interest such as moments with a convenient reference measure which is approximated by an numerical transport, leading to a perturbed prior. Subsequently, a coordinate transformation leads to a beneficial setting for the further function approximation. An efficient layer based transport construction is realized by using the Variational Monte Carlo (VMC) method. The convergence analysis covers all terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback-Leibler divergence. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a central motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity.

• D. Hömberg, R. Lasarzik, Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness, Preprint no. 2671, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2671 .
  Abstract, PDF (338 kByte)
  In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions.

• A. Rathsfeld, On a half-space radiation condition, Preprint no. 2669, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2669 .
  Abstract, PDF (341 kByte)
  For the Dirichlet problem of the Helmholtz equation over the half space or rough surfaces, a radiation condition is needed to guarantee a unique solution, which is physically meaningful. If the Dirichlet data is a general bounded continuous function, then the well-established Sommerfeld radiation condition, the angular spectrum representation, and the upward propagating radiation condition do not apply or require restrictions on the data, in order to define the involved integrals. In this paper a new condition based on a representation of the second derivative of the solution is proposed. The twice differentiable half-space Green's function is integrable and the corresponding radiation condition applies to general bounded functions. The condition is checked for special functions like plane waves and point source solution. Moreover, the Dirichlet problem for the half plane is discussed. Note that such a “continuous” radiation condition is helpful e.g. if finite sections of the rough-surface problem are analyzed.

• R. Lasarzik, Maximal dissipative solutions for incompressible fluid dynamics, Preprint no. 2666, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2666 .
  Abstract, PDF (225 kByte)
  We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.

• M. Ebeling-Rump, D. Hömberg, R. Lasarzik, Th. Petzold, Topology optimization subject to additive manufacturing constraints, Preprint no. 2629, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2629 .
  Abstract, PDF (1099 kByte)
  In Topology Optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg-Landau term. During 3D Printing overhangs lead to instabilities, which have only been tackled unsatisfactorily. The novel idea is to incorporate an Additive Manufacturing Constraint into the phase field method. A rigorous analysis proves the existence of a solution and leads to first order necessary optimality conditions. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the Additive Manufacturing Constraint brings about support structures, which can be fine tuned according to engineering demands. Stability during 3D Printing is assured, which solves a common Additive Manufacturing problem.

• M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, Preprint no. 2626, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2626 .
  Abstract, PDF (424 kByte)
  In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

• F. Auricchio, E. Bonetti, M. Carraturo, D. Hömberg, A. Reali, E. Rocca, Structural multiscale topology optimization with stress constraint for additive manufacturing, Preprint no. 2615, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2615 .
  Abstract, PDF (2815 kByte)
  In this paper a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraint and potentially multiple materials or multiscales is analyzed. First order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible workflow to obtain a 3D-printed object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer.

• R. Lasarzik, E. Rocca, G. Schimperna, Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model, Preprint no. 2608, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2608 .
  Abstract, PDF (338 kByte)
  In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.

• J.I. Asperheim, P. Das, B. Grande, D. Hömberg, Th. Petzold, Numerical simulation of high-frequency induction welding in longitudinal welded tubes, Preprint no. 2600, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2600 .
  Abstract, PDF (3294 kByte)
  In the present paper the high-frequency induction (HFI) welding process is studied numerically. The mathematical model comprises a harmonic vector potential formulation of the Maxwell equations and a quasi-static, convection dominated heat equation coupled through the joule heat term and nonlinear constitutive relations. Its main novelties are twofold: A new analytic approach permits to compute a spatially varying feed velocity depending on the angle of the Vee-opening and additional spring-back effects. Moreover, a numerical stabilization approach for the finite element discretization allows to consider realistic weld-line speeds and thus a fairly comprehensive three-dimensional simulation of the tube welding process.

• M. Eigel, L. Grasedyck, R. Gruhlke, D. Moser, Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations, Preprint no. 2580, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2580 .
  Abstract, PDF (1235 kByte)
  We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length.

• M. Drieschner, M. Eigel, R. Gruhlke, D. Hömberg, Y. Petryna, Comparison of monomorphic and polymorphic approaches for uncertainty quantification with experimental investigations, Preprint no. 2579, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2579 .
  Abstract, PDF (6838 kByte)
  Unavoidable uncertainties due to natural variability, inaccuracies, imperfections or lack of knowledge are always present in real world problems. To take them into account within a numerical simulation, the probability, possibility or fuzzy set theory as well as a combination of these are potentially usable for the description and quantification of uncertainties. In this work, different monomorphic and polymorphic uncertainty models are applied on linear elastic structures with non-periodic perforations in order to analyze the individual usefulness and expressiveness. The first principal stress is used as an indicator for structural failure which is evaluated and classified. In addition to classical sampling methods, a surrogate model based on artificial neural networks is presented. With regard to accuracy, efficiency and resulting numerical predictions, all methods are compared and assessed with respect to the added value. Real experiments of perforated plates under uniaxial tension are validated with the help of the different uncertainty models.

Talks, Poster

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

• N. Alia, M.J. Arenas Jaén, Revealing secrets of industrial processes with Math, Lange Nacht der Wissenschaften (Long Night of the Sciences) 2019, WIAS at Leibniz Headquarters, Berlin, June 15, 2019.

• M.J. Arenas Jaén, Modeling, simulation and optimization of inductive pre- and post-heating for thermal cutting of steel plates, Workshop on Mathematics and Materials Science for Steel Production and Manufacturing, June 4 - 5, 2019, Thon Hotel Høyers, Skien, Norway, June 5, 2019.

• C. Brée, V. Raab, D. Gailevičius, V. Purlys, J. Montiel, G.G. Werner, K. Staliunas, A. Rathsfeld, U. Bandelow, M. Radziunas, Genetically optimized photonic crystal for spatial filtering of reinjection into broad-area diode lasers, CLEO/Europe-EQEC 2019, Munich, June 23 - 27, 2019.

• R. Gruhlke, Annual report 2019 -- MuScaBlaDes (subproject 4 within SPP1886), Jahrestreffen des SPP 1886, September 26 - 27, 2019, TU Hamburg-Harburg, September 26, 2019.

• R. Gruhlke, Bayesian upscaling with application to failure analysis of adhesive bonds in rotor blades, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6--II ``Uncertainty Computations with Reduced Order Models and Low-Rank Representations'', June 24 - 26, 2019, Crete, Greece, June 24, 2019.

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

• H. Heitsch, On probabilistic capacity maximization in a stationary gas network, 9th International Congress on Industrial and Applied Mathematics (ICIAM), Minisymposium MS A6-1-1 4 ``Mathematical Optimization and Gas Transport Networks: Academic Developments II'', July 15 - 19, 2019, Valencia, Spain, July 16, 2019.

• H. Heitsch, Optimal Neumann boundary control of the vibrating string with uncertain initial data and probabilistic terminal constraints, The XV International Conference on Stochastic Programming (ICSP XV), Minisymposium ``Nonlinear Programming with Probability Functions'', July 29 - August 2, 2019, Norwegian University of Science and Technology, Trondheim, Norway, July 30, 2019.

• M. Ebeling-Rump, Topology optimization subject to additive manufacturing constraints, INdAM Workshop MACH2019 ``Mathematical Modeling and Analysis of Degradation and Restoration in Cultural Heritage'', March 25 - 29, 2019, Istituto Nazionale di Alta Matematica ``Francesco Severi'', Rome, Italy, March 26, 2019.

• M. Ebeling-Rump, Topology optimization subject to additive manufacturing constraints, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Infinite-Dimensional Optimization of Nonlinear Systems (Part III)'', August 5 - 8, 2019, Berlin, August 6, 2019.

• M. Eigel, A machine learning approach for explicit Bayesian inversion, Workshop 3 within the Special Semester on Optimization ``Optimization and Inversion under Uncertainty'', November 11 - 15, 2019, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, November 12, 2019.

• M. Eigel, A statistical learning approach for high-dimensional PDEs, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6--IV ``Uncertainty Computations with Reduced Order Models and Low-Rank Representations'', June 24 - 26, 2019, Crete, Greece, June 25, 2019.

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

• M. Eigel, A statistical learning approach for parametric PDEs, École Polytechnique Fédérale de Lausanne (EPFL), Scientific Computing and Uncertainty Quantification, Lausanne, Switzerland, May 14, 2019.

• M. Eigel, Some thoughts on adaptive stochastic Galerkin FEM, Sixteenth Conference on the Mathematics of Finite Elements and Applications (MAFELAP 2019), Minisymposium 17 ``Finite Element Methods for Efficient Uncertainty Quantification'', June 18 - 21, 2019, Brunel University London, Uxbridge, UK, June 18, 2019.

• R. Henrion, Chance constraints then and now, International Conference on Stochastic Optimization and Related Topics, April 25 - 26, 2019, Mühlheim an der Ruhr, April 26, 2019.

• R. Henrion, Nonsmoothness in the context of probability functions, Workshop 4 within the Special Semester on Optimization ``Nonsmooth Optimization'', November 25 - 27, 2019, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, November 25, 2019.

• R. Henrion, On derivatives of probability functions, Workshop ``Statistics, Risk & Optimization'', Universität Wien, Austria, September 27, 2019.

• R. Henrion, On some extended models of chance constraints, Workshop ``Mathematical Optimization of Systems Impacted by Rare, High-Impact Random Events'', June 24 - 28, 2019, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, USA, June 24, 2019.

• R. Henrion, Optimal Neumann boundary control of the vibrating string under random initial conditions, OVA9: 9th International Seminar on Optimization and Variational Analysis, Universidad Miguel Hernández, Elche, Spain, September 2, 2019.

• R. Henrion, Optimal Neumann boundary control of the vibrating string with uncertain initial data, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``PDE-constrained Optimization under Uncertainty (Part I)'', August 5 - 8, 2019, Berlin, August 8, 2019.

• R. Henrion, Optimal probabilistic control of the vibrating string under random initial conditions, PGMO DAYS 2018, Session 1E ``Stochastic Optimal Control'', December 3 - 4, 2019, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF'Lab Paris-Saclay, Palaiseau, France, December 4, 2019.

• R. Henrion, Optimization problems with probust constraints: Theory, applications and algorithmic solution, XXXVIII Spanish Conference on Statistics and Operations Research, September 3 - 6, 2019, Universitat Politècnica de València, Alcoi, Spain, September 5, 2019.

• R. Henrion, Probabilistic constraints in optimization with PDEs, Workshop 3 within the Special Semester on Optimization ``Optimization and Inversion under Uncertainty'', November 11 - 15, 2019, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, November 13, 2019.

• R. Henrion, Problèmes d'optimisation sous contraintes en probabilité, Spring School in Nonsmooth Analysis and Optimization, April 16 - 18, 2019, Université Mohammed V, Rabat, Morocco.

• R. Henrion, Robust control of a sweeping process with probabilistic end-point constraints, The XV International Conference on Stochastic Programming (ICSP XV), Minisymposium ``Nonlinear Programming with Probability Functions'', July 29 - August 2, 2019, Norwegian University of Science and Technology, Trondheim, Norway, July 30, 2019.

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

• D. Hömberg, European collaboration in industrial and applied mathematics, EMMC International Workshop 2019 ``European Materials Modelling Council'', Vienna, Austria, February 25 - 27, 2019.

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

• D. Hömberg, MSO for steel production and manufacturing, 9th International Congress on Industrial and Applied Mathematics (ICIAM), Minisymposium IM FT-4-2 1 ``Academia-Industry Case Studies from MI-NET and ECMI'', July 15 - 19, 2019, Valencia, Spain, July 15, 2019.

• D. Hömberg, Mathematics for steel production and manufacturing, Conference ``Dynamics, Equations and Applications (DEA 2019)'', Session ``D442 Complex Systems in Material Science'', September 16 - 20, 2019, AGH University of Science and Technology, Kraków, Poland, September 17, 2019.

• D. Hömberg, Mathematics for steel production and manufacturing, Sondierungsworkshop MPIE/WIAS ``Elektrochemie, Halbleiternanostrukturen und Metalle'', October 14 - 15, 2019, Max-Planck-Institut für Eisenforschung GmbH Düsseldorf, October 14, 2019.

• D. Hömberg, Maths for digital factory, Polytechnic of Leiria, Center for Rapid and Sustainable Product Development, Marinha Grande, Portugal, October 10, 2019.

• D. Hömberg, Optimal pre-heating strategies for the flame cutting of steel plates, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Infinite-Dimensional Optimization of Nonlinear Systems (Part III)'', August 5 - 8, 2019, Berlin, August 6, 2019.

• O. Klein, On uncertainty quantification for models involving hysteresis effects, Seminar Nichtlineare Optimierung und Inverse Probleme, WIAS, Berlin, May 21, 2019.

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

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

• R. Lasarzik, Weak entropy solutions in the context of induction hardening, 9th International Congress on Industrial and Applied Mathematics (ICIAM), Minisymposium CP FT-1-7 9 ``Partial Differential Equations VII'', July 15 - 19, 2019, Valencia, Spain, July 19, 2019.

• R. Lasarzik, Weak solutions and weak-strong uniqueness to a thermodynamically consistent model describing solid-liquid phase transitions, WIAS Workshop ``PDE 2019: Partial Differential Equations in Fluids and Solids'', September 9 - 13, 2019, Berlin, September 12, 2019.

• M. Marschall, Adaptive low-rank approximation in Bayesian inverse problems, 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Minisymposium 6--IV ``Uncertainty Computations with Reduced Order Models and Low-Rank Representations'', June 24 - 26, 2019, Crete, Greece, June 25, 2019.

• M. Marschall, Complexity reduction in Bayesian inverse problems by low-rank tensor representation, Robert Bosch GmbH, Corporate Research -- Advanced Engineering Computer Vision Systems (CR/AEC4), Hildesheim, April 16, 2019.

• M. Marschall, Low-rank surrogates in Bayesian inverse problems, 19th French-German-Swiss Conference on Optimization (FGS'2019), Minisymposium 1 ``Recent Trends in Nonlinear Optimization 1'', September 17 - 20, 2019, Nice, France, September 17, 2019.

• M. Marschall, Random domains in PDE problems with low-rank surrogates. Forward and backward, Physikalisch-Technische Bundesanstalt, Arbeitsgruppe 8.41 ``Mathematische Modellierung und Datenanalyse'', Berlin, April 10, 2019.