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Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Sommersemester 2014
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
Di 15.04.14
16:15
MA 313
Vu Hoang Linh
(Vietnam National U Hanoi)
Spectral analysis of differential-algebraic equations (Abstract)
V. Mehrmann
Di 22.04.14
16:15
MA 313
Bernhard Bodmann
(U Houston)
Near-optimal robust transmissions with random fusion frames (Abstract)
J. Vybiral
Di 29.04.14
16:15
MA 313
Alexander Linke (WIAS Berlin)
On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime (Abstract)
G. Bärwolff
Di 06.05.14
16:15
MA 313
tba
tba
R. Schneider
Di 13.05.14
16:15
MA 313
Hongguo Xu
(U of Kansas, Lawrence)
A classical Poisson-Nernst-Planck model for ionic flow
V. Mehrmann
Di 20.05.14
16:15
MA 313
Elias Wegert
(TU Freiberg)
to be announced
J. Liesen
Di 27.05.14
16:15
MA 313
Martin Lotz
(U Manchester)
Phase Transitions in Convex Optimization (Abstract)
G. Kutyniok
Di 03.06.14
16:15
MA 313
Michael Hinze
(U Hamburg)
to be announced
G. Bärwolff
Di 10.06.14
16:15
MA 313
Lucy Weggler
(HU Berlin)
to be announced
K. Schmidt
R. Schneider
Di 24.06.14
16:15
MA 313
Martin Ehler
(U Wien)
to be announced
B. Bodmann
Di 01.07.14
16:15
MA 313
Kees Vuik
(TU Delft)
to be announced
R. Nabben
Di 08.07.14
16:15
MA 313
Yoel Shkolnisky
(Tel Aviv U)
Viewing Directions Estimation in cryo-EM Using Synchronization (Abstract)
G. Kutyniok
Di 22.07.14
16:15
MA 313
Raymond Chan
(Chinese U Hong Kong)
Point-spread function reconstruction in ground-based astronomy (Abstract)
G. Kutyniok
Di 20.01.15
16:15
MA 313
Peter Casazza
(U of Missouri)
to be announced
G. Kutyniok

Abstracts zu den Vorträgen:

Vu Hoang Linh (Vietnam National U Hanoi)

Spectral analysis of differential-algebraic equations
Dienstag, den 15.04.2014, 16.15 Uhr in MA 313

Abstract:
The spectral theory of differential equations started with the historical works of Lyapunov and Perron. This is an important part of the qualitative theory of differential equations and it gives a powerful tool for analyzing the stability and the asymptotic behavior of solutions. For ordinary differential equations, while the theoretical part of the spectral analysis (for both the finite and the infinite dimensional cases) was fairly well established by late 70's, the numerical aspects have attracted researchers' attention since 80's. In particular, in the last fifteen years, a significant progress has happened with the numerical methods for approximating spectral intervals of differential and differential-algebraic equations. In this talk, we first summarize our recent contributions to the theory and to the numerical methods for the spectral analysis of differential-algebraic equations. Then, we discuss also works in progress and several open problems that are worth being investigated in the next coming years.

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

Bernhard Bodmann (U Houston)

Near-optimal robust transmissions with random fusion frames
Dienstag, den 22.04.2014, 16.15 Uhr in MA 313

Abstract:
This talk describes recent results on fusion frames which are motivated by communication theory. Fusion frames are families of weighted orthogonal projection operators which sum to an approximate identity. Fusion frames are a natural tool in packet-based communication systems when a vector to be transmitted is encoded in terms of lower-dimensional components obtained by projecting the vector onto subspaces. The possible non-orthogonality between the subspaces can be used to correct errors that occur in the transmission. This talk discusses the problem of fusion frame design for the recovery from partial data loss when the content of a relatively small number of subspace components is lost in the transmission. Optimal fusion frames are in general hard to find, but randomized constructions give near-optimal fusion frames in a straightforward way. Essential techniques for frame design in this context come from the literature on compressed sensing.

Alexander Linke (WIAS Berlin)

On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime
Dienstag, den 29.04.2014, 16.15 Uhr in MA 313

Abstract:
Unfortunately, nearly all mixed discretizations for the incompressible Navier-Stokes equations do not preserve exactly the fundamental identity of vector calculus `gradient fields are irrotational’. In consequence, mixed methods suffer from a non-physical interaction of divergence-free and irrotational forces in the momentum balance, resulting in the well-known numerical instability of `poor mass conservation'. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields.

Therefore, a new variational crime for the nonconforming Crouzeix-Raviart element is proposed, where divergence-free, lowest-order Raviart-Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct an efficient flow discretization for unstructured and even anisotropic 2D and 3D simplex meshes that fulfills `gradient fields are irrotational'. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

Martin Lotz (U Manchester)

Phase Transitions in Convex Optimization
Dienstag, den 27.05.2014, 16.15 Uhr in MA 313

Abstract:
Convex regularization has become a popular approach to solve large scale inverse or data separation problems. A prominent example is the problem of identifying a sparse signal from linear samples my minimizing the l_1 norm under linear constraints. Recent empirical research indicates that many convex regularization problems on random data exhibit a phase transition phenomenon: the probability of successfully recovering a signal changes abruptly from zero to one as the number of constraints increases past a certain threshold. We present a rigorous analysis that explains why phase transitions are ubiquitous in convex optimization. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems, to demixing problems, and to cone programs with random affine constraints. These applications depend on a new summary parameter, the statistical dimension of cones, that canonically extends the dimension of a linear subspace to the class of convex cones.
Joint work with Dennis Amelunxen, Mike McCoy and Joel Tropp.

Yoel Shkolnisky (Tel Aviv U)

Viewing Directions Estimation in cryo-EM Using Synchronization
Dienstag, den 8.07.2014, 16.15 Uhr in MA 313

Abstract:
A central stage in recovering the structure of large proteins (3D density maps) from their 2D cryo-electron microscopy (cryo-EM) images, is to determine a three-dimensional model of the protein given many of its 2D projection images. The direction from which each image was taken is unknown, and the images are small and extremely noisy. The goal is to determine the direction from which each image was taken, and then to combine the images into a three-dimensional model of the molecule.

We present an algorithm for determining the viewing directions of all cryo-EM images at once, which is robust to extreme levels of noise. The algorithm is based on formulating the problem as a synchronization problem, that is, we estimate the relative spatial configuration of pairs of images, and then estimate a global assignment of orientations that satisfies all pairwise relations. Information about the spatial relation of pairs of images is extracted from common lines between triplets of images. These noisy pairwise relations are combined into a single consistent orientations assignment, by constructing a matrix whose entries encode the pairwise relations. This matrix is shown to have rank 3, and its non-trivial eigenspace is shown to reveal the projection orientation of each image. In particular, we show that the non-trivial eigenvectors encode the rotation matrix that corresponds to each image.

No prior knowledge is required.

This is a joint work with Amit Singer from Princeton University.

Raymond Chan (Chinese U Hong Kong)

Point-spread function reconstruction in ground-based astronomy
Dienstag, den 22.07.2014, 16.15 Uhr in MA 313

Abstract:
Ground-based astronomy refers to acquiring images of objects in outer space via ground-based telescopes. Because of atmospheric turbulence, images so acquired are blurry. One way to estimate the unknown blur or point spread function (PSF) is by using natural or artificial guide stars. Once the PSF is known, the images can be deblurred using well-known deblurring methods.
Another way to estimate the PSF is to make use the aberration of wavefronts received at the telescope, i.e., the phase, to derive the PSF. However, the phase is not readily available; instead only its low-resolution gradients can be collected by wavefront sensors. In this talk, we will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and then the PSF in high accuracy.
Our model can be solved efficiently by alternating direction method of multiplier whose convergence has been well established. Numerical results will be given to illustrate that our new model is efficient and give more accurate estimation for the PSF.

Zusatzinformationen / Extras