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- ItemLow rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Eigel, Martin; Grasedyck, Lars; Gruhlke, Robert; Moser, DieterWe 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.
- ItemEffective diffusion in thin structures via generalized gradient systems and EDP-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Frenzel, Thomas; Liero, MatthiasThe notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin--de Donder kinetics.
- ItemA unified analysis of algebraic flux correction schemes for convection–diffusion equations(Berlin ; Heidelberg : Springer, 2018) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr; Rankin, RichardRecent results on the numerical analysis of algebraic flux correction (AFC) finite element schemes for scalar convection–diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.
- ItemThe sharp-interface limit for the Navier--Stokes--Korteweg equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Abels, Helmut; Daube, Johannes; Kraus, Christiane; Kröner, DietmarWe investigate the sharp-interface limit for the Navier--Stokes--Korteweg model, which is an extension of the compressible Navier--Stokes equations. By means of compactness arguments, we show that solutions of the Navier--Stokes--Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions.
- ItemOn the numerical range of sectorial forms(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Linke, Alexander; Rehberg, JoachimWe provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H∞-angle for the H∞-calculus on Lp for all p ∈ (1, ∞) if the coefficients are real valued.
- ItemLower large deviations for geometric functionals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Hirsch, Christian; Jahnel, Benedikt; Tóbiás, AndrásThis work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.
- ItemSelf-consistent field theory for a polymer brush. Part I: Asymptotic analysis in the strong-stretching limit(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Münch, Andreas; Wagner, BarbaraIn this study we consider the self-consistent field theory for a dry, in- compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the self-consistent field theory in the strong-stretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strong-stretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the self-consistent field theory.
- ItemSimulation and design of a compact GaAs based tunable dual-wavelength diode laser system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Koester, Jan-Philipp; Radziunas, Mindaugas; Zeghuzi, Anissa; Wenzel, Hans; Knigge, AndreaWe present our design of a compact, integrated and tunable dual-wavelength diode laser system emitting around 785 nm, which is of interest for several applications like Raman spectroscopy and the generation of THz radiation. To achieve a more compact device compared to previous GaAs based designs two etch depths are realized, leading to shallowly etched ridge waveguides in regions were optical gain is applied and deeply etched waveguides used to enable compact integrated waveguide components. The device parameters are optimized using a numerically efficient simulation tool for passive waveguides. Subsequently, the entire laser system is further analyzed applying a sophisticated traveling-wave equation based model for active devices giving access to internal intensity and carrier density distributions. It is shown that active laser simulations are crucial to deduce critical and performance limiting design aspects not accessible via an all-passive simulation.
- ItemSimulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Gräßle, Carmen; Hintermüller, Michael; Hinze, Michael; Keil, TobiasWe are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.
- ItemSelf-consistent field theory for a polymer brush. Part II: The effective chemical potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Münch, Andreas; Wagner, BarbaraThe most successful mean-field model to describe the collective behaviour of the large class of macromolecular polymers is the self-consistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the mean-field equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strong-stretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching.