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Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Sommersemester 2015
Verantwortliche Dozenten:
Alle Professoren der
Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen
Koordination:
Dr. Christian Schröder, Dr. Hans-Christian Kreusler
Termine:
Di 16-18 Uhr in MA 313 und nach Vereinbarung
Inhalt:
Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen

Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Terminplanung / schedule
Datum
date
Zeit
time
Raum
room
Vortragende(r)
speaker
Titel
title
Einladender
invited by
20.10.15
16:15
MA 313
Michael Hinze
(U Hamburg)
Ehrenkolloquium aus Anlaß des 65. Geburtstags von Günter Bärwolff:
Simulation und Kontrolle von (Mehrphasen-)Strömungen (Abstract)
F. Tröltzsch
10.11.15
16:15
MA 313
Wolfgang Wendland
(U Stuttgart)
On the Gauss minimal energy problem with Riesz potentials (Abstract (PDF, 166,4 KB))
F. Tröltzsch
17.11.15
16:15
MA 313
Kees Vuik
(TU Delft)
Coupled preconditioners for the Incompressible Navier Stokes Equations (Abstract)
R. Nabben
1.12.15
16:15
MA 313
Patrick Joly
(INRIA Saclay)
Time Domain Perfectly Matched Layers for electromagnetic wave propagation in dispersive media (Abstract)
K. Schmidt
5.01.16
16:15
MA 313
Carola-Bibiane Schönlieb
(U Cambridge)
What is the right sparsity for imaging? (Abstract)
G. Kutyniok
19.01.16
16:15
MA 313
Philipp Grohs
(ETH Zürich)
Directional representation systems in numerics (Abstrakt)
G. Kutyniok
26.01.16
16:15
MA 313
Lutz Tobiska
(U Magdeburg)
Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations (Abstrakt)
E. Emmrich

Abstracts zu den Vorträgen:

Michael Hinze (U Hamburg)

Simulation und Kontrolle von (Mehrphasen-)Strömungen
Dienstag, den 20.10.2015, 16.15 Uhr in MA 313
Abstract

Im Rahmen meines Vortrags werde ich aktuelle Entwicklungen bei der Simulation und Kontrolle von Zweiphasenströmungen mit Phasenfeldmodellen skizzieren. Im ersten Teil des Vortrags werde ich ein praktisches adaptives Verfahren für die Simulation vorstellen, das auftretende Fehler sowohl in der Strömung als auch im Phasenfeld zuverlässig und effizient schätzt. Im 2ten Teil des Vortrags werde ich die Kontrolle von (Zweiphasen-)Strömungen diskutieren, wobei der Schwerpunkt auf closed-loop Konzepten liegen wird. Die Konzepte werden anhand zahlreicher Beispiele illustriert.

Diese Arbeiten wurden in Teilen gemeinsam mit den Kollegen Harald Garcke (Regensburg), Michael Hintermüller (HU Berlin) und Christian Kahle (Hamburg) durchgeführt.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Kees Vuik (TU Delft)

Coupled preconditioners for the Incompressible Navier Stokes Equations
Dienstag, den 17.11.2015, 16.15 Uhr in MA 313
Abstract

After linearization and discretization of the incompressible Navier Stokes equations one has to solve block-structured indefinite linear systems. The successful design of robust, scalable, and efficient preconditioners is intimately connected with an understanding of the structure of the resulting block matrix system. Effective preconditioners are often based on an approximate block decomposition of the discretized incompressible Navier Stokes equations. This requires a careful consideration of the spectral properties of the component block operators and their Schur complement operators. Through this purely algebraic view of preconditioning, a simplified system of block component equations is developed. Inclusion of "physics based" preconditioners of the various parts can lead to effective preconditioners with optimal or nearly optimal convergence rates for academic and industrial problems.

CFD applications in maritime industry, for example hull resistance prediction, involve high Reynolds number flows modelled by the incompressible Reynolds-averaged Navier-Stokes equations. The system of equations is discretized with a cell-centered finite-volume method with colocated variables. After linearization, various SIMPLE-type preconditioners can be applied to solve the discrete system. In this presentation, we discuss their performance for flows with Reynolds number up to 10^9 and cell aspect ratio up to 10^6.

Finally we note that the Augmented Lagrangian preconditioner proposed by Benzi and Olshanskii works well for a finite element discretization of the Oseen equation. We have generalized the method to finite volume discretization of the Navier-Stokes equations. Some theoretical and numerical results are given.

This is joint work with Xin He, and Chris Klaij.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Patrick Joly (INRIA Saclay)

Time Domain Perfectly Matched Layers for electromagnetic wave propagation in dispersive media
Dienstag, den 1.12.2015, 16.15 Uhr in MA 313
Abstract

This work analyses the Perfectly Matched Layers (PMLs) for a large class of dispersive media, particularly for Negative Index Metamaterials (NIMs). Unfortunately, classical PMLs lead to instabilities due to the presence of backward waves, whose phase and group velocities have opposite signs. This is a well-known result for non-dispersive media. In the more general case of dispersive media, which include NIMs, we perform an analysis of a large class of PMLs (which includes classical PMLs) and deduce a necessary and sufficient stability criterion. We use this criterion to show the inherent instability of classical PMLs in NIMs and to propose new stable PMLs in this case. We illustrate numerically for the Drude model the instability of the classical PMLs and the stability of the new ones.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Carola-Bibiane Schönlieb (U Cambridge)

What is the right sparsity in imaging?
Dienstag, den 5.01.2016, 16.15 Uhr in MA 313
Abstract

When assigned with the task of reconstructing an image from imperfect data the first challenge one faces is the derivation of a truthful image and data model. In the context of sparse reconstructions, some of this task amounts to selecting the right basis in which the image has a sparse representation. This can be determined by the a-priori knowledge about the image, the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we optimise our model choice? Starting from the first modelling strategy this talk will lead us from the total variation as one of the most successful sparse regularisation models for images today to non-smooth second- and third-order regularisers, combined with data models for Gaussian and Poisson distributed data as well as impulse noise. Applications for image denoising, inpainting and surface reconstruction are given. After a critical discussion of these different image and data models we will turn towards the second modelling strategy and propose to combine it with the first one using a bilevel optimization method. In particular, we will consider optimal parameter derivation for total variation denoising with multiple noise distributions and optimising total generalised variation regularisation for its application in photography.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Philipp Grohs (ETH Zürich)

Directional representation systems in numerics
Dienstag, den 19.01.2016, 16.15 Uhr in MA 313
Abstract

Directional representation systems such as curvelets, shearlets or ridgelets have made a big impact in image- and signal processing in the last decades, due to their superior ability of resolving curved singularities in multivariate functions, arising for instance as edges in image data. Despite their approximation properties (which are vastly superior over standard discretizations such as wavelets for FEM for the approximation of functions with curved singularities) the use of directional representation systems in numerical analysis is still at its infancy. In this talk we discuss the useability of directional representation systems in numerical analysis, namely for the numerical solution of different types of PDEs whose generic solutions possess curved singularities. Concretely, we present a ridgelet-based adaptive solver for linear advection equations which can be shown to converge optimally, even for solutions with line singularities, as well as a construction of shearlet systems on bounded domains which may be used for the design of optimally convergent adaptive solvers for elliptic PDEs whose solutions possess singularities along curved submanifolds. The results are based on joint work with Simon Etter, Gitta Kutyniok, Jackie Ma, Axel Obermeier and Philipp Petersen.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Lutz Tobiska (U Magdeburg)

Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations
Dienstag, den 26.01.2016, 16.15 Uhr in MA 313
Abstract

Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H^1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H^1 and L^2 errors of the velocity and the L^2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results. This talk is based on joint work with Alexander Linke, Weierstraß-Institut, Berlin and Gunar Matthies, Institut für Numerische Mathematik, Technische Universität Dresden.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

Zusatzinformationen / Extras