Our research seminar addresses graduate students, young researchers, and well-established experts interested in the area of numerical analysis, optimization, and scientific computing. We aim to discuss recent developments in our field, as well as Ph.D. and Master's theses.

Prof. Dr. Patrick Henning
Prof. Dr. Katharina Kormann
Jun.-Prof. Dr. Markus Weimar

Summer term 2022

Tue, 23.08.2022 (13:00 in IB 1/103; Friedrich-Sommer-Raum):
BENJAMIN DÖRICH (Karlsruhe Institut für Technologie (KIT))
Maximum norm error bounds for the full discretization of non-autonomous wave equations

In this talk, we discuss the full discretization of a non-autonomous wave equation in Omega by isoparametric finite elements in space and the implicit Euler method in time. Here, Omega is a convex bounded domain with a regular boundary. Building upon the work of Baker and Dougalis [1], we prove maximum norm estimates for the semi discretization in space and the full discretization. The key tool in the analysis is to trade in integrability, coming from the inverse of the discretized differential operator, for time derivatives on the error in the H1 x L2-norm. We show how the ideas of the semi discretization transfer to the fully discrete error analysis, yielding a good starting point for higher order methods in time and the treatment of quasilinear problems.

[1] G. A. Baker and V. A. Dougalis, On the L-convergence of Galerkin approximations for second-order hyperbolic equations, Math. Comp. 34 (1980), no. 150, pp. 401–424. URL https://doi.org/10.2307/2006093
[2] B. Dörich, J. Leibold and B. Maier, Maximum norm error bounds for the full discretization of non-autonomous wave equations. CRC 1173 Preprint 2021/47, Karlsruhe Institute of Technology, 2021. URL https://www.waves.kit.edu/downloads/CRC1173_Preprint_2021-47.pdf

Thu, 11.08.2022 (13:00 in IB 1/103; Friedrich-Sommer-Raum):
The connection between interpolation spaces and approximation spaces for non-linear approximation

Approximation theory gives results about how functions can be best approximated by simpler functions. A central question in that context is the characterization of the set of all functions with the same order of approximation by a given approximation method. In this talk, we will use results from interpolation theory to formulate assumptions under which this characterization problem can be solved. For that, we use an equivalence relation between approximation and interpolation spaces, proven by DeVore and Popov, to establish very concrete approximation spaces for certain non-linear approximations. Specifically, we will establish approximation spaces for certain free knot spline approximations by using Lebesgue spaces and Besov spaces.

Wed, 25.05.2022 (14:00 15:00 in IB 1/103; Friedrich-Sommer-Raum):
Rate-optimal sparse approximation of compact break-of-scale embeddings

Approximation theory tells us that the expected rate of convergence of numerical methods is closely related to the regularity of the object we want to approximate. Besides classical isotropic Sobolev smoothness, the notion of dominating mixed regularity of functions turned out to be an important concept in numerical analysis. Although the theory of embeddings within those scales seems to be well-understood, not that much is known about break-of-scale embeddings. In the talk we define new function spaces of Besov- and Triebel-Lizorkin-type with hybrid smoothness that include both scales of function spaces discussed above. We present some embeddings and construct explicit (non-)linear algorithms based on hyperbolic wavelets that yield sharp dimension-independent rates of convergence. The talk is based on recent joint work [1] with Glenn Byrenheid and Markus Weimar.

[1] G. Byrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approximation of compact break-of-scale embeddings, preprint arXiv:2203.10011, (2022).

Thu, 28.04.2022 (12:15 in IA 1/53):
Introduction to discrepancy theory and the Koksma-Hlawka inequality

Let f : [0,1]d -> IR be an integrable function on the unit cube in dimension d. We want to approximate its integral using a finite set of sampling points. This is because the dimension might be very high and the function f might be 'complicated' in the sense that we cannot calculate the integral exactly using analytical methods. When thinking of numerical approximation of an integral, one of the first things which come to mind are quadrature rules. Hence, at first, we will have a look on their definition. We then will introduce some measure for a 'good' distribution for the nodes of our quadrature rule, namely the 'discrepancy', and we will see that the approximation error depends on this. Moreover, we will have a look on a special class of integrands, namely 'functions with bounded variation'. We find some upper bounds for the error of integration which only depend on the variation of the integrand and the discrepancy of our nodes. This result is known as the 'Koksma-Hlawka Inequality’. Finally, I will give a brief outlook on polynomial lattice point sets which might have low discrepancy and are therefore interesting in terms of 'good' nodes for our quadrature formula.

Winter term 2021 / 2022

Guest: Johan Wärnegård (KTH Stockholm) will visit our group from March 08 - 11, 2022.

Thu, 10.03.2022 (13:00 in IB 1/103; Friedrich-Sommer-Raum):
JOHAN WÄRNEGÅRD (KTH Royal Institute of Technology, Stockholm)
Energy-conservative finite element methods for nonlinear Schrödinger equations

The cubic nonlinear Schrödinger equation, also known as the Gross-Pitaevskii equation, has applications in many fields. In this talk I will introduce the equation and show how, using a spatial discretization based on the method of Localized Orthogonal Decomposition, it is possible to approximate the initial value with 6th order accuracy, O(H^6), with respect to the chosen mesh size H, measured in terms of the time invariants of the equation. Perhaps surprisingly this only assumes H^4(D)-regularity of the initial value, which is compatible with physically relevant problems. I shall also demonstrate the dramatic effect inaccurate representation of the time invariants can have on the numerical solution. Finally, this superconvergent space-discretization is combined with an efficient Crank-Nicolson time integrator tailored for the LOD-space so that the initial low errors are preserved. Significant computational gains are achieved and illustrated by means of numerical examples.

Thu, 03.03.2022 (14:15, Zoom - please contact Markus Weimar for details):
MARC HOVEMANN (Philipps-University Marburg)
Triebel-Lizorkin-Morrey Spaces and Differences: Characterizations and Applications

In this talk our subject of interest are the Triebel-Lizorkin-Morrey spaces Esu,p,q(IRd). They have been introduced in 2005 by Tang and Xu as a generalization of the original Triebel-Lizorkin spaces Fsp,q(IRd). In the first part of the talk we will give a precise definition of the Triebel-Lizorkin-Morrey spaces. For that purpose we use the Fourier analytical approach and work with a smooth dyadic decomposition of the unity. Moreover we collect some basic properties of the spaces Esu,p,q(IRd). In the second chapter we present characterizations in terms of differences for the Triebel-Lizorkin-Morrey spaces. Here we show so-called ball-mean characterizations. For the proofs of those results we used some conditions concerning the parameters s, u, p and q. We will see that some of these conditions are also necessary. In the third part of the talk we will discuss several applications of our characterization in terms of differences. So on the one hand we can use differences to study the properties of special test functions in connection with the spaces Esu,p,q(IRd). On the other hand differences can be applied to investigate the behavior of Truncation operators in the context of Triebel-Lizorkin-Morrey spaces. One of those operators is given by (Tf)(x)=|f(x)| where we have f in Esu,p,q(IRd). It turns out that under some conditions on the parameters the operator T is bounded on Triebel-Lizorkin-Morrey spaces. At the end of the talk we want to mention further applications of characterizations by differences. So for example they also can be used to prove new Quarklet characterizations for the Triebel-Lizorkin-Morrey spaces.

Thu, 03.02.2022 (14:15 in IB 1/103; Friedrich-Sommer-Raum):
MATTHIAS S. MAIER (Texas A&M University)
Optical Phenomena, Resonances, and Homogenization of Layered Heterostructures

Two-dimensional materials with controllable electronic structures have opened up an unprecedented wealth of optical phenomena that challenge the classical picture of electromagnetic wave propagation. They are a promising ingredient in the design of novel optical devices, promising "subwavelength optics" beyond the diffraction limit.

In this talk we present a homogenization theory of plasmonic crystals with two-dimensional material inclusions and a spectral analysis quantitatively describing Lorentz resonances in the effective permittivity tensor. We conclude by demonstrating how our analytical findings can serve as an efficient computational tool to describe the general frequency dependence of periodic optical devices.

Guest: Glenn Byrenheid (FSU Jena) visited our group from January 20 - 22, 2022.

Fri, 21.01.2022 (14:15 in IB 1/103; Friedrich-Sommer-Raum):
Faber-Schauder meets mixed smoothness

In 1909 G. Faber proved that every continuous function can be expanded into a series of hat functions. The interesting point: the corresponding coefficients are generated by point evaluations. This fact together with the easy piecewise linear structure established it as a proven object in numerical analysis.

H. Triebel employed a tensorization of this system to characterize multivariate function spaces with mixed smoothness. This approach allows us an easy but effective introduction into sampling theory of the corresponding function spaces.

We will give an overview of applications in high dimensional approximation theory, supplement in an easy but effective way to the picture of Lq([0, 1]) and H1([0, 1]) (sampling) approximation, discuss optimality, and complement these results by best m-term approximation with respect to the Faber-Schauder dictionary.

Thu, 13.01.2022 (14:15 in IB 1/103; Friedrich-Sommer-Raum):
Spectral coarse spaces for overlapping Schwarz methods based on energy-minimizing extensions

Discretizing partial differential equations often results in sparse linear equation systems, and high spatial resolutions lead to large systems, which can be solved efficiently using iterative methods. A suitable class of solvers are Krylov methods preconditioned by domain decomposition preconditioners, which are scalable and robust for a wide range of problems. Unfortunately, highly heterogeneous problems arising, e.g., in the simulation of composite materials or porous media generally lead to unfavorable distributions of the eigenvalues of the system matrix that cause slow convergence for many solvers, including classical domain decomposition preconditioners.

In order to retain robustness of domain decomposition methods, the coarse space can be enriched by additional coarse basis functions computed from eigenmodes of local generalized eigenvalue problems, leading to so-called spectral or adaptive coarse spaces. This talk deals with a specific class of adaptive coarse spaces for overlapping Schwarz methods which are based on a partition of the interface of the corresponding nonoverlapping domain decomposition. In particular, the generalized eigenvalue problems are based on energy-minimizing extensions corresponding to the interface components. The presented methods have a provable condition number bound, which is independent of the contrast of the coefficient functions.

A great challenge is to construct adaptive coarse spaces that are robust but can be built algebraically, that is, using only the fully assembled system matrix without additional Neumann matrices or geometrical information. In this talk, a novel adaptive coarse space that is both robust and algebraic is introduced. Furthermore, approaches for combining adaptive coarse spaces with nonlinear preconditioning as well as machine learning techniques are discussed.

The talk is based on joint work with Axel Klawonn, Jascha Knepper, Martin Lanser, Janine Weber (University of Cologne), Oliver Rheinbach (TU Bergakademie Freiberg), Kathrin Smetana (Stevens Institute of Technology), and Olof Widlund (New York University).

Thu, 04.11.2021 18.11.2021 (14:15 in IB 1/103; Friedrich-Sommer-Raum):
Stability of traveling oscillating fronts in Ginzburg Landau equations

In complex Ginzburg Landau equations many different wave phenomena occur. There are special solutions which maintain their shape while traveling in space and oscillating in the complex plane. One specific class of such solutions are traveling oscillating fronts (TOFs). Their profile decays at minus infinity but approaches a nonzero limit at plus infinity. In this talk we give results on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at plus infinity. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at infinity. Using slowly decaying exponential weights, the framework allows to derive appropriate resolvent estimates, semi-group techniques, and Gronwall estimates.

Joint work with Wolf-Jürgen Beyn (Bielefeld University)

Summer term 2021

Mon, 16.08.2021 (15:00, Zoom):
Tree approximation in Besov- and Triebel-Lizorkin-type spaces based on wavelet expansions

The talk is concerned with a non-linear approximation method via tree approximations based on wavelet expansions. We derive the exact rate of convergence of the error of the best N-term tree approximation w.r.t. embeddings between Besov and Triebel-Lizorkin spaces in all possible combinations. Finally a comparison is drawn between the rates of convergence of the well-known unconstrained best N-term, the linear non-adaptive and the tree approximation method on open and bounded domains.

Joint work with Markus Weimar

Winter term 2020 / 2021

Mon, 01.02.2021 (10:00, Zoom-Meeting-ID: 981 3709 7351, Password: 643923):
CHRISTIAN KREUZER (TU Dortmund University)
Oscillation in a posteriori error analysis

A posteriori error estimators are a key tool for the quality assessment of given finite element approximations to an unknown PDE solution as well as for the application of adaptive techniques.

Typically, the estimators are equivalent to the error up to an additive term, the so called oscillation. It is a common believe that this is the price for the `computability' of the estimator and that the oscillation is of higher order than the error. Cohen, DeVore, and Nochetto [1], however, presented an example, where the error vanishes with the generic optimal rate, but the oscillation does not. Interestingly, in this example, the local $H^{-1}$-norms are assumed to be computed exactly and thus the computability of the estimator cannot be the reason for the asymptotic overestimation. In particular, this proves both believes wrong in general.

In this talk, we present a new approach to posteriori error analysis, where the oscillation is dominated by the error. The crucial step is a new splitting of the data into oscillation and oscillation free data. Moreover, the estimator is computable if the discrete linear system can essentially be assembled exactly. After presenting the fundamental principles for linear elements, we shall discuss extensions to higher order methods.

[1] A. Cohen, R. DeVore, and R. H. Nochetto, Convergence Rates of AFEM with $H^{-1}$ Data, Found. Comput. Math. 12 (2012):671–718

Joint work with Andreas Veeser (Universita degli Studi di Milano, Italy)

Summer term 2020

Guests: Cornelia Schneider (FAU Erlangen-Nürnberg) and Petru A. Cioica-Licht (University of Duisburg-Essen) visited our group from September 28 - 30, 2020.

Winter term 2019 / 2020

Wed, 18.03.2020 (14:30 in IB 1/103; Friedrich-Sommer-Raum):
CORNELIA SCHNEIDER (Friedrich-Alexander University of Erlangen-Nürnberg)
Sobolev and Besov regularity of parabolic PDEs

The talk is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in a specific scale of Besov spaces. It is known that in many cases the order of convergence of adaptive wavelet-schemes depends on the regularity of the solution in these Besov spaces. On the other hand it is the fractional Sobolev regularity which determines the rate of convergence of non adaptive (uniform) algorithms. Therefore, in order to justify the use of adaptive schemes for solving parabolic PDEs, an analysis of the regularity of the solution in the scale of Besov spaces and a comparison with its Sobolev regularity is needed. It turns out that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

Winter term 2018 / 2019

Thu, 29.11.2018 (14:30 in IB 1/103; Friedrich-Sommer-Raum):
MARKUS HANSEN (Philipps-University Marburg)
Properties of Kondratiev spaces and Besov regularity for semilinear elliptic PDEs on polyhedral domains

Kondratiev spaces are a special type of weighted Sobolev spaces, particularly suited to describe the regularity of solutions to operator equations on polygonal or polyhedral domains.
The relevance of these spaces from the point of view of numerical analysis stems from embedding assertions into Besov spaces, which allows to derive convergence rates for adaptive wavelet and Finite element methods.
In this talk we shall discuss properties of these spaces, starting with the motivation from regularity theory, followed by structural properties (embeddings, localization principles) and pointwise multiplication assertions. Finally, all those results are combined to derive new regularity results for semilinear elliptic equations.

Winter term 2017 / 2018

Wed, 22.11.2017 (16:15 in NA 2/24):
LARS DIENING (Bielefeld University)
Linearization of the p-Poisson equation

This is a joint work with Massimo Fornasier and Maximilian Wank. In this talk we propose a iterative method to solve the non-linear p-Poisson equation. The method is derived from a relaxed energy by an alternating direction method. We are able to show algebraic convergence of the iterates to the solution. However, our numerical experiments based on finite elements indicate optimal, exponential convergence.

Summer term 2017

Tue, 22.08.2017 (13:15 in NA 2/64):
Nichtlineare Approximationsraten und Besov-Regularität elliptischer PDEs auf Polyedergebieten

Wed, 26.07.2017 (16:15 in NA 2/64):
PETRU A. CIOICA-LICHT (University of Otago, Dunedin, New Zealand)
Stochastic Partial Differential Equations: Regularity and Approximation

Stochastic partial differential equations (SPDEs, for short) are mathematical models for evolutions in space and time, which are corrupted by noise. Although we can prove existence and uniqueness of a solution to various classes of such equations, in general, we do not have an explicit representation of this solution. Thus, in order to make those models ready to use for applications, we need efficient numerical methods for approximating their solutions. And to determine the efficiency of an approximation method, we usually need to analyse the regularity of the target object, which is, in our case, the solution of the SPDE.
The aim of this talk is to present some recent results concerning the regularity of SPDEs and to point out their relevance for the question of developing efficient numerical methods for solving these equations. Prior to that I will give a brief overview over that parts of the already established Lp-theory for SPDEs that is relevant in this context. For simplicity, we focus on the most basic example, the stochastic heat equation driven by a (cylindrical) Wiener process.

Mon, 26.06.2017 (14:15 in NA 3/24):
ROB STEVENSON (University of Amsterdam, Netherlands)
Adaptive wavelet methods for space-time variational formulations of evolutionary PDEs

Space-time discretization methods require a well-posed space-time variational formulation. Such formulations are well-known for parabolic problems. The (Navier)-Stokes equations can be viewed as a parabolic problem for the divergence-free velocities. Yet to avoid the cumbersome construction of divergence-free trial spaces, we present well-posed variational formulations for the saddle-point problem involving the pair of velocities and pressure. We discuss adaptive wavelet methods for the optimal adaptive solution of simultaneous space-time variational formulations of evolutionary PDEs. Thanks to use of tensor products of temporal and spatial wavelets, the whole time evolution problem can be solved at a complexity of solving one instance of the corresponding stationary problem.

Wed, 21.06.2017 (16:15 in NA 01/99):
Adaptive Wavelet-Methoden für Operatorgleichungen

Diese Antrittsvorlesung gibt einen Einblick in die Grundlagen moderner Methoden der numerischen Analysis zur approximativen Lösung von Operatorgleichungen, wie sie in einer Vielzahl von Modellen naturwissenschaftlich-technischer Disziplinen auftreten. Im Zentrum der Diskussion steht dabei die Beschreibung und Analyse der sogenannten adaptiven Wavelet-Galerkin-Methode. Im Zuge dessen werden insbesondere die Grundlagen adaptiver Verfahren und die damit verbundene Regularitätstheorie in Funktionenräumen vom Sobolev- und Besov-Typ vorgestellt, sowie elementare Konzepte der nichtlinearen Approximationstheorie besprochen.

Wed, 19.04.2017 (14:15 in NA 2/64):
MAX GUNZBURGER (Florida State University, Tallahassee, USA)
Integral equation modeling for nonlocal diffusion and mechanics

We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

Mon, 27.03.2017 (14:15 in NA 2/64):
SUSANNE C. BRENNER (Louisiana State University, Baton Rouge, USA)
C0 Interior Penalty Methods

C0 interior penalty methods are discontinuous Galerkin methods for fourth order problems that are based on standard Lagrange finite element spaces for second order problems. In this talk we will discuss the a priori and a posteriori error analyses of these methods for fourth order elliptic boundary value problems and elliptic variational inequalities.