Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

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.

Rückblick

• Kolloquium ModNumDiff WS 2012/13 [2]
• Kolloquium ModNumDiff SS 2012 [3]
• Kolloquium ModNumDiff WS 2011/12 [4]
• Kolloquium ModNumDiff SS 2011 [5]
• Kolloquium ModNumDiff WS 2009/10 [6]
• Kolloquium ModNumDiff SS 2009 [7]
• Kolloquium ModNumDiff WS 2008 [8]
• Kolloquium ModNumDiff SS 2008 [9]
• Kolloquium ModNumDiff WS 2007 [10]
• Kolloquium ModNumDiff SS 2007 [11]
• Kolloquium ModNumDiff WS 2006/07 [12]
• Kolloquium ModNumDiff SS 2006 [13]
• Kolloquium ModNumDiff WS 2005/06 [14]
• Kolloquium ModNumDiff SS 2005 [15]

Abstracts zu den Vorträgen:

Erwan Faou (INRIA Rennes & ENS Paris)

Hamiltonian splitting for Vlasov equations and long time behavior problems
Dienstag, den 16.04.2013, 16.15 Uhr in MA 313
Abstract:

We consider Vlasov equations and first discuss their geometrical structures and long time behavior properties. We will then consider their numerical discretization by semi-Lagrangian methods. We will show how it is possible to build high order in time methods that preserve the Hamiltonian structure of the equation. We will discuss convergence results, and also show that for Vlasov-Poisson equations, specific properties of the system allow to reduce the number of space interpolation to be done at each step. This is a joint work with F. Casas, N. Crouseilles and M. Mehrenberger.

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

Anders Hansen (U Cambridge)

This talk had to be cancelled!
From spectra to sampling - Functional analysis meets computational harmonic analysis
Dienstag, den 14.05.2013, 16.15 Uhr in MA 313
Abstract:

In this talk I will discuss how recent developments in computational spectral theory has had impact on modern sampling theory and compressed sensing. In particular, new techniques from computational spectral theory allow for substantial generalisations of the state of the art sampling and compressed sensing methods and theory. I will also discuss new results on how to break the so called coherence barrier in compressed sensing. More precisely, it is well known that compressed sensing relies on two key pillars: incoherence and sparsity. Although there are examples where one has both of these features, there are surprisingly many cases in applications where these criteria are not satisfied, in particular Fourier sampling and wavelet or polynomial reconstruction (both essential in Magnetic Resonance Imaging (MRI)). This has led to a substantial gap between the state of the art compressed sensing theory and its actual use in application.

I will show how to bridge this fundamental gap by introducing a new theory where the assumptions of incoherence and sparsity are replaced by two new features: asymptotic incoherence and asymptotic sparsity as well as multi-level sampling. With these new tools at hand I will demonstrate two rather intriguing effects: The success of compressed sensing is resolution dependent and optimal sampling strategies are signal structure dependent.

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

Ulrich Langer (U Linz)

BEM-based FEM: A Non-Standard Finite Element Method based on Boundary Integral Operators
Dienstag, den 28.05.2013, 16.15 Uhr in MA 313
Abstract:

We present and analyze a new non-standard finite element method based on element-local boundary integral operators that permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDE-harmonic trial functions and can thus be interpreted as a local Trefftz method. In fact, in his classical paper, Trefftz (1926) already proposed to use PDE-harmonic trial or basis functions in a domain decomposition framework. Our construction principle requires the explicit knowledge of the fundamental solution of the partial differential operator, but only locally, in every polyhedral element. This allows us to solve PDEs with elementwise constant coefficients. This construction technique via boundary integral operator representations of the Poincarè - Steklov operator is borrowed from the boundary element domain decomposition method proposed by Wendland and Hsiao (1991) and then used by many authors.

In this talk we mainly consider the diffusion equation as a model problem, but the method can be generalized to convection-diffusion-reaction problems and to systems of PDEs like the linear elasticity system and the time-harmonic Maxwell equations with elementwise constant or, at least, elementwise smooth coefficients. We provide a rigorous H^1- and L_2-error analysis of the method for smooth and non-smooth solutions under quite general assumptions on the geometric properties of the polyhedral elements in the case of the diffusion equation. It turns out that, for the convection-diffusion problems, the exact version (without approximation of the fluxes) of our scheme is the best possible SUPG scheme in the sense as discussed by Brezzi, Franca, Hughes and Russo (1997), Hughes, Feijo, Mazzei and Quincy (1998), and Hughes and Sangalli (2007a). Finally, we discuss fast solvers for these non-standard finite element schemes. Numerical results confirm our theoretical predictions.

This talk is based on a joint work with Clemens Hofreither and Clemens Pechstein within the Doctoral Program "Computational Mathematics" supported by the Austrian Science Fund FWF under the grant W1214, project DK4. Earlier results were obtained in joint work with Dylan Copeland and David Pusch within the FWF project P19255.

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

Felix Krahmer (U Göttingen)

Sigma-Delta quantization for compressed sensing
Dienstag, den 11.06.2013, 16.15 Uhr in MA 313
Abstract:

Compressed sensing is a novel paradigm in signal processing concerning the recovery of approximately sparse signals from undersampled linear measurements. Recently, this approach has been combined with coarse quantization allowing the reconstruction of the signal from quantized Gaussian measurements with polynomial accuracy.

In the talk, we will discuss a new approach for the error analysis of such schemes linking them to the Restricted Isometry Property and present generalizations of the result to more general classes of measurements.

This is joint work with Joe-Mei Feng, Rayan Saab, and Özgür Yılmaz.

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

Arjan van der Schaft (U Groningen)

Structure-preserving model reduction of port-Hamiltonian systems
Dienstag, den 25.06.2013, 16.15 Uhr in MA 313
Abstract:

Port-Hamiltonian models arise from first principles network modeling of multi-physics systems. They are useful for analysis and control since they reflect the underlying physical characteristics of the systems. In case of large-scale systems, or in case of systems arising from structure-preserving discretization of port-Hamiltonian distributed parameter systems, there is a clear need for model reduction methods which preserve the port-Hamiltonian structure of the models.

After giving a general introduction to port-Hamiltonian systems theory we will outline in this talk a number of structure-preserving model reduction approaches for (linear) port-Hamiltonian systems, illustrate them on some examples, and discuss open problems.

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