Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Rückblick

• Kolloquium ModNumDiff WS 2011/12
• Kolloquium ModNumDiff SS 2011
• Kolloquium ModNumDiff WS 2009/10
• Kolloquium ModNumDiff SS 2009
• Kolloquium ModNumDiff WS 2008
• Kolloquium ModNumDiff SS 2008
• Kolloquium ModNumDiff WS 2007
• Kolloquium ModNumDiff SS 2007
• Kolloquium ModNumDiff WS 2006/07
• Kolloquium ModNumDiff SS 2006
• Kolloquium ModNumDiff WS 2005/06
• Kolloquium ModNumDiff SS 2005

Abstracts zu den Vorträgen:

Cesar Palencia (U Valladolid)

Runge-Kutta convolution quadrature for abstract, linear, homogeneous Volterra equations
Dienstag, den 10.04.2012, 16.15 Uhr in MA 313
Abstract:

Recent results concerning Runge-Kutta based convolution quadrature methods for abstract, well posed, linear, and homogeneous Volterra equations show a general representation of the numerical solution in terms of the continuous one. The interest of such a representation goes beyond the error and stability analysis, since it explains that the numerical solution, under suitable methods, inherits some important qualitative properties, such as positivity or contractivity, of the exact solution. In this talk we will summarize this theory and use it for the construction of fully discrete, transparent boundary conditions, for the heat and Schroedinger equation, with good qualitative properties.

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

Massimo Fornasier (TU Muenchen)

Analysis and Simulation of Particle Systems and Kinetic Equations Modeling Interacting Agents in High Dimension
Dienstag, den 17.04.2012, 16.15 Uhr in MA 313
Abstract:

We are addressing the analysis and the tractable simulation of dynamical systems which are modeling the behavior in a social context of a large number N of complex interacting agents described by a large amount of parameter (high-dimension).We are facing the following fundamental challenges and by describing them in detail we are also pointing to new research directions:

- Random projections and recovery for high-dimensional dynamical systems: we shall explore how concepts of data compression via Johnson-Lindenstrauss random embeddings onto lower-dimensional spaces can be applied for tractable description and simulation of complex dynamical interactions. As a fundamental subtask for the recovery of high-dimensional trajectories from low-dimensional simulated ones, we will address the efficient recovery of point clouds defined on embedded manifolds from random projections.

- Mean field equations: for the limit of the number N of agents to infinity, we shall further explore how the concepts of compression can be generalized to work for associated mean field equations.

- Approximating functions in high-dimension: differently from purely physical problems, in the real life the social forces which are ruling the dynamics are actually not known. Hence we will address the problem of automatic learning from collected data the fundamental functions governing the dynamics.

- Sparse optimal control in high-dimension and mean field optimal control: while self-organization of such dynamical systems has been so far a mainstream, we will focus on their sparse optimal control, modeling best policies. We will investigate L1-minimization to design sparse optimal controls.

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

Moritz Kaßmann (U Bielefeld)

Regularity theory for parabolic nonlocal operators
Dienstag, den 24.04.2012, 16.15 Uhr in MA 313
Abstract:

We extend Moser's works from 1964, 1967 and 1971 to nonlocal problems, i.e. we prove Hölder-regularity and a Harnack-type inequality for weak solutions to parabolic equations with integro-differential operators of fractional order. Assumptions on the kernel are discussed in detail and several examples are provided. This work is joint with M. Felsinger.

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

Vladimir Sergeichuk (National Academy Ukraine, Kiev)

Classification problems for systems of forms and linear mappings
Dienstag, den 8.05.2012, 16.15 Uhr in MA 313
Abstract:

A method that reduces the problem of classifying systems of bilinear/sesquilinear forms and linear mappings to the problem of classifying systems of linear mappings is presented.

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

Janosch Rieger (U Frankfurt a.M.)

Galerkin Finite Elements for Partial Differential Inclusions
Dienstag, den 22.05.2012, 16.15 Uhr in MA 313
Abstract:

The partial differential inclusions considered in this talk arise in a natural way from the study of constrained control problems and deterministic uncertainty. In this setting, the solution set contains all solutions induced by admissible controls or perturbations. We currently try to develop the necessary analytical background and numerical methods for an efficient approximation of this solution set.

First experiments in the linear elliptic case show that it is difficult to obtain good results by discretizing the multivalued right-hand side $F$. It is much better to project the differential inclusion to some finite-dimensional space and approximate the solution set of the resulting algebraic inclusion.

The semi-linear elliptic case is much more involved. Set-valued Nemytskii operators have to be considered for the projection of the inclusion to a finite element space. In order to guarantee uniform convergence of the Galerkin solution sets, the so-called \emph{relaxed one-sided Lipschitz property} is imposed on the right-hand side $F$, which is a generalization of the classical OSL property. Under this assumption, it is also possible to discretize and approximate the unknown Galerkin solution sets.

Work on semi-linear parabolic inclusions is still in progress, but almost completed. The convergence of the Galerkin solution sets has been established. In this case, the Galerkin inclusions are ordinary differential inclusions, and their reachable sets can (at least theoretically) be computed by availabe numerical methods.

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

Kasso Okoudjou (U of Maryland)

Probabilistic frames
Dienstag, den 29.05.2012, 16.15 Uhr in MA 313
Abstract:

In this talk I will give an overview of finite frame theory from a probabilistic point of view. In particular, I will review the notion of probabilistic frames and indicate how it is a natural generalization of frames. Concepts such as tight frames, frame potential have their natural counter parts in the setting of probabilistic frames. I will describe these new concepts and will show how probabilistic frames appear in other areas such as statistics, convex geometry, the theory of spherical design, and quantum computing. The talk is based on recent joint work with Martin Ehler.

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

Zouwei Shen (National U Singapore)

MRA based wavelet frame and applications
Dienstag, den 5.06.2012, 17.00 Uhr in MA 313
Abstract:

One of the major driving forces in the area of applied and computational harmonic analysis during the last two decades is the development and the analysis of redundant systems that produce sparse approximations for classes of functions of interest. Such redundant systems include wavelet frames, ridgelets, curvelets and shearlets, to name a few. This talk focuses on tight wavelet frames that are derived from multiresolution analysis and their applications in imaging. The pillar of this theory is the unitary extension principle and its various generalizations, hence we will first give a brief survey on the development of extension principles. The extension principles allow for systematic constructions of wavelet frames that can be tailored to, and effectively used in, various problems in imaging science. We will discuss some of these applications of wavelet frames. The discussion will include frame-based image analysis and restorations, image inpainting, image denoising, image deblurring and blind deblurring, image decomposition, segmentation and CT image reconstruction.

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

Emmanuel Candes (Stanford U)

Compressive Sensing (Matheonvortrag)
Mittwoch, den 20.06.2012, 18.00 Uhr in MA 005
Abstract:

Compressive sensing is a new sampling / data acquisition theory based on the discovery that one can exploit sparsity or compressibility when acquiring signals of general interest, and that one can design nonadaptive sampling techniques that condense the information in a compressible signal into a small amount of data. Interestingly, this may be changing the way engineers think about signal acquisition in areas ranging from analog-to-digital conversion, digital optics, magnetic resonance imaging, and seismics. This talk will introduce fundamental conceptual ideas underlying this new sampling or sensing theory. There are already many ongoing efforts to build a new generation of sensing devices based on compressive sensing, and we will address remarkable recent progress in this area as well.

Philipp Grohs (ETH Zuerich)

Algorithmic Treatment of Nonlinear Data
Dienstag, den 3.07.2012, 16.15 Uhr in MA 313
Abstract:

One of the characteristic features of our modern age is the deluge of data we are confronted with. In addition to the mere masses, we are witnessing more and more modern sensing devices where measurements are of a nonstandard form with data points constrained to nonlinear geometries. With applications ranging from topics in human biomechanics, over image processing, kinematics and robotics to computer graphics, nonlinear geometric data represents a fact of life in modern computational science and therefore it is of eminent interest to develop computational and theoretical tools capable of processing it in an efficient manner. Unfortunately, due to the nonlinear structure inherent in geometric data, classical linear methods known from signal processing, such as wavelet transforms, finite elements, etc., cannot be used in a meaningful way. This calls for genuinely new constructions which respect the underlying geometric structure while satisfying the same desirable properties of well-known linear methods. In my talk I will discuss several such constructions.

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

Stephan Dahlke (U Marburg)

The Continuous Shearlet Transform in Arbitrary Space Dimension: General Setting, Shearlet Coorbit Spaces, Traces, and Embeddings
Dienstag, den 10.07.2012, 16.15 Uhr in MA 313
Abstract:

One of the most important tasks in modern applied mathematics is the analysis of signals. One particular problem which is currently in the center of interest is the extraction of directional information, and several different approaches have been suggested. Among all these approaches, the shearlet transform stands out because it is related to group theory, i.e., it can be derived from a square-integrable representation of a certain group, the full shearlet group.

In the first part of this talk, we are concerned with the generalization of this concept to arbitrary space dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations, and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces, the shearlet coorbit spaces, and associated Banach frames can be derived.

In the second part of the talk, we discuss the structural properties of shearlet coorbit spaces. We show that compactly supported functions with enough smoothness and vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. The results are based on general atomic decompositions of Besov spaces. Furthermore, we will discuss trace results for these subspaces with respect to the coordinate planes. It turns out that in many cases these traces are again contained in lower dimensional shearlet coorbit spaces.

This is joint work with G. Steidl (Kaiserslautern) and G. Teschke (Neubrandendburg).

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