# Trimester Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45, Bonn (unless stated otherwise)

Organizers: Susanne C. Brenner, Björn Engquist, Max Gunzburger, Daniel Peterseim, Marc Alexander Schweitzer

## Friday, January 6

14:15 - 15:00 Christian Stohrer: Finite Element Heterogeneous Multiscale Methods for Maxwell's Equations

Abstract: To approximate the effective behavior an electromagnetic wave propagating through a multiscale medium, we adapt the Finite Element Heterogeneous Multiscale Method to Maxwell's equations. The proposed method can be applied to time-harmonic and time dependent Maxwell's equations. In the talk we show how a priori-error estimates can be proven using the notion of T-coercivity for time-harmonic Maxwell's equations and a Strang-type lemma for the time-dependent case.

15:00 - 16:00 Michael Feischl: Fast random field generation with H-matrices

Abstract: We use the H-matrix technology to compute the approximate square root of a covariance matrix in linear complexity. This allows us to generate normal and log-normal random fields on general finite point sets with optimal complexity. We derive rigorous error estimates, which show that particularly for short correlation length, this new method outperforms the standard method of truncating the Karhunen-Loève expansion.

Besides getting rid of the log-factor in the complexity estimate, the method requires only mild assumptions on the covariance function and on the point set. Therefore, it might also be an alternative to circulant embedding, which is well-defined only for regular grids and stationary covariance functions.

## Tuesday, January 17

15:00 - 16:00 Chi-Wang Shu: Discontinuous Galerkin Finite Element Method for Multiscale Problems

Abstract: In this talk, we first give a brief introduction to the discontinuous Galerkin method, which is a finite element method using completely discontinuous basis functions, for solving hyperbolic conservation laws and parabolic and elliptic equations. We will then survey the progress in developing discontinuous Galerkin methods for multiscale problems, in three different approaches, namely using the heterogeneous multiscale method (HMM) framework, using domain decompositions, and using multiscale basis in the discontinuous Galerkin method. Numerical results will be shown to demonstrate the effectiveness of the multiscale discontinuous Galerkin methods.

## Wednesday, January 18

15:00 - 16:00 Alexandre Ern: Finite element quasi-interpolation and best approximation

Abstract: We introduce quasi-interpolation operators for scalar- and vector-valued finite element spaces with some continuity across mesh interfaces. These operators are stable in L1, are projections, and deliver optimal local approximation estimates in Sobolev spaces. The theory is illustrated on H1-, H(curl)- and H(div)-conforming finite element spaces.

## Thursday, January 19

15:00 - 16:00 Robert Scheichl: Multilevel Quasi Monte Carlo Methods for Uncertainty Quantification

Abstract: Large-scale PDE-constrained uncertainty propagation and Bayesian inference are inherently difficult and computationally intensive problems. Sampling methods are among the most accurate and promising methods for these problems. Their cost is dimension independent, thus rendering them very suitable for the infinite-dimensional PDE setting. However, classical sampling approaches, such as Monte Carlo methods or Metropolis-Hastings MCMC, are very slow to converge and often infeasible for realistic applications. The problem becomes even more involved when the differential operator depends in a non-affine way on the random parameters and when the quantities of interest depend nonlinearly on the PDE solution. A popular model problem for this situation is the lognormal diffusion problem which is of actual practical interest in hydrology. Quasi-Monte Carlo methods and their multilevel extensions provide alternatives with potentially vastly improved computational efficiencies, in terms of cost to accuracy. We provide a dimension-independent convergence analysis for the lognormal problem for the forward uncertainty propagation case, as well as for the inverse Bayesian inference setting supported by numerical experiments.

## Tuesday, January 24

15:00 - 16:00 Donald L. Brown: Multiscale Methods and Stability in Some High-Frequency Helmholtz-Type Problems

Abstract: In this talk, we will discuss issues in the computation of high-frequency Helmholtz-type problems. In particular, we discuss the issue of pollution effects and how certain multiscale LOD correction methods can eliminate the effect in certain resolution regimes. This will help to motivate the issue of frequency dependent stability in continuous problems. We then present a few of the current results on frequency explicit bounds for heterogenous acoustic and elastic media.

## Thursday, January 26

14:30 - 16:00 Wolfgang Hackbusch: Numerical Tensor Calculus

## Tuesday, January 31

15:00 - 16:00 Ivan Graham: Fast UQ for diffusion and neutron transport

Abstract: This is joint work with F. Kuo and I.H. Sloan (UNSW), D. Nuyens (Leuven) and M. Parkinson and R. Scheichl (Bath).

We shall describe some practical uncertainty quantification problems involving the diffusion equation and the Boltzmann transport equation where the uncertainty is described by stationary Gaussian random fields. Small length scale and high variance presents PDEs problems with a challenging multiscale structure. We describe a UQ algorithm for these applications which combines circulant embedding techniques for the sampling with Quasi Monte-Carlo methods for computing the required high-dimensional integrals. The method is capable of handling very high stochastic dimension and is consistently faster than Monte-Carlo methods. Multilevel variants can be used to obtain further acceleration. We describe some recent theory and computations.

## Thursday, February 2

15:00 - 16:00 Ivan Oseledets: Topological optimization and aposteriori estimates: is it possible to compute the constant?

Abstract: Joint work with George Ovchinnikov, Vlad Pimanov, Denis Zorin.

A major problem in topology optimization in thermal or elasticity problems is the emergence of so-called checkerboard structures, where the minimum of the discretized FEM functional does not correspond to a nice value of the original functional. Or it maybe, and the true solution is a certain microstructure. A typical approach is either to do filtering of the topology to remove bad configurations, or introduce penalty terms for them (for example, by penalizing the length of the boundary of the obtaned shape). We propose a fresh view on this problem, based on a classical connection between the exact solution and the solution of the discretized version, which leads to a computable upper bound of the functional. By minimizing this upper bound we get a guarantee for the effectiveness of the computed topology. Another challenge is that to get a final number, we do not only need the aposteriori error estimate, but we need the exact value of the constant. We provide a series of different solutions obtained by the optimization algorithm for different empirical values of the constant, but we are not yet able to numerically compute the true desired values of the functionals for such structures, thus leaving the main question about the necessity of the microstructures in the optimal topology still open.

## Tuesday, February 21

15:00 - 16:00 Sören Bartels: Stable discretization of singular flows and application to a problem in optimal insulation

## Thursday, March 2

15:00 - 16:00 Viet Ha Hoang: A Hierarchical Finite Element Monte Carlo Method for Stochastic Two Scale Elliptic Problems

Abstract: We consider two scale elliptic equations whose coefficient is random and depends on a macroscopic slow variable and a fast variable. We assume that the effective coefficient can be approximated by solving random cell problems in a finite size cube (this is the case, for example, of an ergodic random coefficient, or a random periodic coefficient). This approximated effective coefficient is, however, realization dependent; and we aim to compute its expectation. Straightforward employment of finite element approximation and the Monte Carlo method to compute this expectation with the same level of finite element resolution and the same number of Monte Carlo samples at every macroscopic point is prohibitively expensive. We develop a hierarchical finite element Monte Carlo algorithm to approximate the effective coefficients at a dense hierarchical network of macroscopic points. The method achieves an optimal level of complexity that is essentially equal to that for computing the effective coefficient at one macroscopic point, with essentially the same accuracy. The levels of accuracy for solving cell problems and for the Monte Carlo approximation are chosen according to the level in the hierarchy that the macroscopic points belong to. Solutions at those points at which the cell problems are solved with high accuracy and the number of samples in the Monte Carlo approximation is high are employed as correctors for the effective coefficient at those points at which the cell problems are solved with lower accuracy and fewer Monte Carlo samples are used.

The method combines the hierarchical finite element method for solving cell problems at a dense network of macroscopic points with the optimal complexity developed in D. L. Brown, Y. Efendiev and V. H. Hoang, Multiscale Model. Simul. 11 (2013), with a hierarchical Monte Carlo algorithm that uses different number of samples at different macroscopic points depending on the level in the hierarchy that the macroscopic points belong to.

This is a joint work with Donald L. Brown (Nottingham University, UK).

## Thursday, March 9

15:00 - 16:00 John Ball: Mathematics of liquid crystals

## Thursday, March 16

15:00 - 16:00 Patrick Henning: Finite element discretizations for nonlinear Schrödinger equations with rough potentials

## Thursday, March 23

15:00 - 16:00 Emmanuil Georgoulis: Some recent results on finite element methods on polygonal and curved elements

Abstract: I will give an overview of some recent results regarding the analysis of finite element methods (mostly of discontinuous Galerkin flavour) posed on meshes involving extremely general polygonal/polyhedral and/or curved element shapes. In particular, I will discuss the development of discontinuous Galerkin methods for general linear elliptic, parabolic and 1st order hyperbolic problems on general polygonal/polyhedral meshes, paying special attention to meshes with element shapes involving multiple scales and/or arbitrary numbers of faces. I will then move to the "limiting" case of almost arbitrary element shapes involving curved elements, with particular application to the derivation of a posteriori error bounds for interface problems with general interface geometries.

## Tuesday, March 28

15:00 - 16:00 Carsten Carstensen: Adaptive eigenvalue computation

## Thursday, March 30

15:00 - 16:00 Gabriel Barrenechea: Upper and lower bounds of eigenvalues

## Monday, April 10

15:00 - 16:00 Joscha Gedicke: A posteriori error analysis for symmetric Arnold-Winther FEM

Abstract: This talk presents a new a posteriori error analysis for the symmetric mixed finite element by Arnold and Winther. In addition we show some higher order postprocessing of the displacements. Based on the postprocessing we develop an a posteriori error estimator for the stress-velocity formulation of the Stokes problem. Numerical examples verify the proven reliability and efficiency of the a posteriori error estimators.

## Tuesday, April 11

15:00 - 16:00 Illia Karabash: Optimal resonances and optical cavities with gradually changing deviations from periodicity

Venue: HIM seminar room (in the basement), Poppelsdorfer Allee 45

Abstract: A leaky cavity (or resonator) is a heterogeneous region of space within which the electromagnetic (or acoustic) field is well confined, but not completely confined.

Minimization of the leakage is experimentally studied in Photonics by hand designs of the cavity's structure based on heuristic understanding of flow of EM energy, by computer optimization of several parameters describing special types of structures, and recently, by lithographic 'full topology optimization' where every pixel of the design pattern is a degree of freedom. These procedures lead to various types of periodic structures with sophisticatedly tailored defects.

Mathematically, the problem is to design the material parameters of the heterogeneous medium in such a way that they satisfy certain fabrication constraints and generate a resonance as close as possible to the real line. The goals of the talk is to describe difficulties arising in numerical optimization experiments and to provide rigorous analytic background for optimization of resonances in 1-D photonic crystals.

## Wednesday, April 12

15:00 - 16:00 Nicole Cusimano: Towards a fractional model of electrical signal propagation in highly heterogeneous tissue

Venue: HIM seminar room (in the basement), Poppelsdorfer Allee 45

## Thursday, April 13

15:00 - 16:00 Houman Owhadi: Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis

Venue: HIM seminar room (in the basement), Poppelsdorfer Allee 45

Abstract: We show how the discovery/design of robust scalable numerical solvers for arbitrary bounded linear operators can, to some degree, be addressed/automated as a Game/Decision Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space B endowed with a quadratic norm ||⋅||, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of u ∈ B, given partial measurements ([φ_{i}, u])_{i ∈ {1, …, m}} with φ_{i} ∈ B*, using relative error in ||⋅||-norm as a loss) is a centered Gaussian field ξ solely determined by the norm ||⋅||, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary gambles/bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm ||⋅|| and induce a multi-resolution decomposition of B that is adapted to the eigensubspaces of the operator defining the norm ||⋅||. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operators (including arbitrary continuous linear bijections from H^{s}_{0}(Ω) to H^{-s}(Ω) or to L^{2}(Ω)) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to O(N polylog N)$ solvers and eigenspace adapted Multiresolution Analysis (resulting in near linear complexity approximation of all eigensubspaces). This is a joint work with Clint Scovel (Caltech) and a preprint is available at https://arxiv.org/abs/1703.10761.

## Thursday, April 20

14:00 - 15:00 Gantumur Tsogtgerel: On convergence of discrete exterior calculus

15:00 - 16:00 Max Gunzburger: Integral equation modeling for diffusion and mechanics