Online workshop "PDE and Numerical Mathematics"

On 28 June 2021, 14:00-18:00, an online workshop "PDE and Numerical Mathematics" is organised by the Mathematics Departments of the Universities of Münster and Twente.

Please contact Mario Ohlberger via  mario.ohlberger@uni-muenster.de if you are interested to join.

Programme
14:00-14:05 Welcome - Bernard Geurts

14:05-14:30 Mario Ohlberberger (Münster): Challenges in Model Order Reduction for parameterized PDEs

Abstract: In the last two decades there has been a tremendous development and success in projection based model order reduction for parameterized systems. Approaches such as the Reduced Basis Method or system theoretic approaches like Balanced Truncation are meanwhile well developed for certain classes of parameterized systems. The major barrier for these methods lies in the fact, that they rely on linear approximation spaces and are thus limited by the convergence rate of the so called Kolmogorov width. In this talk we will address challenges that result from this observation and sketch possible approaches to overcome this limitation.

14:30-14:55 Matthias Schlottbom (Twente): A model reduction approach for inverse problems with operator valued data

Abstract: We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The highdimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator, which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we develop a novel model reduction approach that takes advantage of the underlying tensor product structure of the problem and which allows to obtain low-dimensional certified reduced order models of quasioptimal rank. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.

15:00-15:25 André Schlichting (Münster): Evolutions equations on graphs and the upwind transportation metric

Abstract: We propose a nonlocal gradient structure approximating the aggregation equation motivated by the classical upwind scheme widely used for the numerical approximation of first order equations. We show, that the nonlocal upwind metric is very well suited for the variational formulation of first-order equations on graphs and graphons. The induced distance is a quasi-metric (non-symmetric) leading to a formal Finslerian structure. We show how the variational framework is suitable to show stability under suitable limits of the graph structure.

15:25-15:50 Erwin Luesink (Twente): Numerical integration of Lie-Poisson equations

Abstract: In this talk I will introduce the concept of a Lie-Poisson equation and how to apply numerical methods designed for integration of differential equations on Lie groups to Lie Poisson equations. Lie-Poisson equations are central in Hamiltonian mechanics with symmetries. Important examples of Lie-Poisson type equations are rigid body dynamics and the Euler equations of an ideal fluid. Careful numerical integration can help preserve the geometric structure that these equations possess.

15:50-16:10 Break

16:10-16:35 Christian Seis (Münster): Stability, mixing and relaxation. Quantitative theories for elementary fluid models

Abstract: I will try to give a rough summary of the topics in my research portfolio. These include stability estimates for linear advection and advection-diffusion equations with applications to fluid mixing, well-posedness and error analyses of numerical schemes, bounds on the heat transport in Rayleigh-Benard convection, and fine large-time asymptotics for nonlinear diffusion equations.

16:35-17:00 James Michael Leahy (Twente): Fluid equations with transport type rough path perturbations

Abstract: We consider the Euler equations for the incompressible flow of an ideal fluid with an additional rough-in-time, divergence-free, Lie-advecting vector field. In recent work, we have demonstrated that this system arises from Clebsch and Hamilton-Pontryagin variational principles with a perturbative geometric rough path Lie-advection constraint. In this paper, we prove local wellposedness of the system and establish a Beale-Kato-Majda (BKM) blow-up criterionIn dimension two, we show that the !?-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.

17:00-17:25 Christian Engwer (Münster): Cut-Cell methods -- decoupling meshes from geometry

Abstract: Cut-Cell methods, like the unfitted FE or unfitted dG method, offer an approach to avoid the difficult step of geometry adapted mesh generation. The actual geometry is intersected with the geometry to construct the actual cut-cell where the solution is approximated. This can lead to arbitrary cuts and arbitrarily small cells. Depending on the actual PDE to solve different stabilization techniques to handle small cells. In general two appraoches are possible; either small cells are merged with larger neighboring cells or penalty methods are used. We will discuss the general ideas of cut-cell methods, present applications and discuss stabilizations methods for different kinds of PDEs.

17:25-17:50 Arnout Franken: Structure-preserving discretization of hyperbolic PDEs

Abstract: Many phenomena in mechanics are modelled using hyperbolic PDEs. These equations often contain a rich mathematical structure, i.e. conservation laws and Hamiltonian structures. In the SPRESTO project, we develop a framework for systematically coarsening nonlinear hyperbolic systems of PDEs with the aim of preserving this structure. The presentation focusses on the Korteweg- de Vries equation, for which different numerical methods are compared in this context.

17:50-17:55 Closing - Mario Ohlberger