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AG Modellierung, Numerik, DifferentialgleichungenKolloquium WS 2013/14

"AG Modellierung, Numerik, Differentialgleichungen"

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Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Wintersemester 2013/14
Verantwortliche Dozenten:
Alle Professoren der
Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen
Koordination:
Dr. Christian Schröder

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 15.10.13
16:15
MA 313
Jens Frehse
(U Bonn)
Zur Existenz- und Regularitätstheorie von Bellmann-Gleichungen (Abstract)
E. Emmrich
Di 22.10.13
16:15
MA 313
Anders Hansen
(U Cambridge)
Compressed sensing in the real world - The need for a new theory (Abstract)
G. Kutyniok
Di 29.10.13
16:15
MA 313
AG ModNumDiff
Arbeitsgruppenbesprechung
F. Tröltzsch
Di 12.11.13
16:15
MA 313
Keith Taylor
(Dalhousie U)
Crystal Groups in Analysis (Abstract)
G. Kutyniok
Di 3.12.13
16:15
MA 313
Remi Gribonval
(INRIA Rennes)
Sparse dictionary learning in the presence of noise and outliers (Abstract)
G. Kutyniok
Mi 8.01.14
18:00
MA 005
Amit Singer
(Princeton U)
Three-dimensional structure determination of molecules without crystallization (Abstract, this is a MATHEON colloquium talk)
G. Kutyniok
Di 4.02.14
16:15
MA 313
Kersten Schmidt
(TU Berlin)
Local impedance boundary conditions for wave propagation in viscous gases (Abstract)
K. Schmidt
Di 11.02.14
16:15
MA 313
Vladimir Kostic
(U Novi Sad)
On matrix nearness problems: distance to delocalization/localization (Abstract)
V. Mehrmann
C. Schröder

Abstracts zu den Vorträgen:

Jens Frehse (U Bonn)

Zur Existenz- und Regularitätstheorie von Bellmann-Gleichungen
Dienstag, den 15.10.2013, 16.15 Uhr in MA 313
Abstract:

Bei Differentialspielen versuchen N Spieler ihre Kostenfunktionale zu minimieren, indem sie ihre Steuerungsfunktionen, über die sie entscheiden, optimal wählen. Die Steuerungen v sind über ein Differentialgleichungssystem gekoppelt. Zum Beispiel könnte das Kostenfunktional des i-ten Spielers die Gestalt

int_0^t (1/2) |v_i(t)|^2 + f_i(x(t)) dt

haben, wobei v und x über ein gewöhnliches Differentialgleichungssystem

d/dt x =g(x,v)

verknüpft sind.

Jeder Spieler versucht sein Kostenfunktional zu minimieren. Die optimale Wahl der Steuerungen lässt sich über das Bellman-System realisieren. Hierzu definiert man nach einem gewissen Formalismus über die Lagrange-Funktion L_i des i-ten Spielers die Hamilton-Funktion H_i(x,p) des i-ten Spielers. In dem obigen Beispiel ist

L_i(x,v,p) = (1/2) |v_i|^2 + f_i(x) + p_i g(x,v).

Man sucht nun nach einem Nash-Gleichgewicht v* der L_i und definiert die Hamilton-Funktion H_i(x,p) = L_i(x,v*(p),p). Das Bellman-System lautet dann

(u_i)_t = H_i(x,grad u)

und wird von der Wertefunktion u=(u_1,...,u_N) der Spieler bzgl. der Anfangsbedingung des gewöhnlichen Differentialgleichungssystems erfüllt. Eine Feedbacksteuerung ist dann v*(grad u(t,x)).

Wird das Differentialgleichungssystem durch Hinzufügen eines weissen Rauschens zu einem stochastischen Differentialgleichungssystem verändert, ergibt sich das Bellman-System in Form einer parabolischen Differentialgleichung (nach Skalierung)

(u_i)_t - Laplace u_i = H_i(x,grad u),

- Laplace u_i + alpha u_i = H_i(x,grad u).

Dieses System wird auf Existenz und Regularität von Lösungen untersucht. Dies impliziert seinerseits die "Lösbarkeit" von Spielen. Stochastische Überlegungen spielen keine Rolle, zum Verständnis des Vortrags wird keinerlei Kenntnis der stochastischen Analysis benötigt.

Insbesondere lassen sich die "stochastischen" Bellmann-Systeme auch aus Nash-Punkten der sogenannten Vlasov-Mc-Kean-Funktionale herleiten; durch diesen Zugang wird bereits in der Problemformulierung auf stochastische Begriffe verzichtet.

Ziel des Vortrag ist es, aufzuzeigen, wieweit die heutige Theorie der partiellen Differentialgleichungen geeignet ist, die bei Differentialspielen auftretenden Bellman-Systeme zu behandeln. Es gibt sehr "vernünftig" aussehende Spiele, die lösbar sein sollten, aber die Theorie der partiellen Differentialgleichungen steht ratlos vor deren Bellman-System. Im "einfachsten" Fall ist das stochastische Differentialgleichungssystem linear in den Steuerungen und die Kostenfunktion quadratisch in denselben. Es lassen sich menschliche Schwächen und Stärken wie Ängstlichkeit, Gemeinheit, Egozentrik und Generosität in den Kostenfunktionalen und der Dynamik modellieren. Die dies modellierenden Terme finden sich in den Hamilton-Funktionen wieder und beeinflussen die Wahl der analytischen Techniken, die man für Regularitätsabschätzungen benötigt, bzw. ermöglichen diese erst oder verhindern sie.

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

Anders Hansen (U Cambridge)

Compressed sensing in the real world - The need for a new theory
Dienstag, den 22.10.2013, 16.15 Uhr in MA 313
Abstract:

Compressed sensing is based on the three pillars: sparsity, incoherence and uniform random subsampling. In addition, the concepts of uniform recovery and the Restricted Isometry Property (RIP) have had a great impact. Intriguingly, in an overwhelming number of inverse problems where compressed sensing is used or can be used (such as MRI, X-ray tomography, Electron microscopy, Reflection seismology etc.) these pillars are absent. Moreover, easy numerical tests reveal that with the successful sampling strategies used in practice one does not observe uniform recovery nor the RIP. In particular, none of the existing theory can explain the success of compressed sensing in a vast area where it is used. In this talk we will demonstrate how real world problems are not sparse, yet asymptotically sparse, coherent, yet asymptotically incoherent, and moreover, that uniform random subsampling yields highly suboptimal results. In addition, we will present easy arguments explaining why uniform recovery and the RIP is not observed in practice. Finally, we will introduce a new theory that aligns with the actual implementation of compressed sensing that is used in applications. This theory is based on asymptotic sparsity, asymptotic incoherence and random sampling with different densities. This theory supports two intriguing phenomena observed in reality: 1. the success of compressed sensing is resolution dependent, 2. the optimal sampling strategy is signal structure dependent. The last point opens up for a whole new area of research, namely the quest for the optimal sampling strategies.

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

Keith Taylor (Dalhousie U)

Crystal Groups in Analysis
Dienstag, den 12.11.2013, 16.15 Uhr in MA 313
Abstract:

Let Iso_d(R) denote the group of isometries of d-dimensional Euclidean space, R^d. A discrete subgroup G of Iso_d(R) such that R^d/G is compact is called a (d-dimensional) crystal group. There are 17 crystal groups in dimension 2 (the wallpaper groups) and 219 in dimension 3. In the 23 problems listed by David Hilbert in 1900, the eighteenth problem asked, among other things, whether the number of crystal groups is finite in each dimension. Bieberbach (1911, 1912) proved that this is the case. Although many algebraic and geometric properties of crystal groups have been well-know for more than a century, they have not played much of a role in analysis until recently. We will talk about the appearance of crystal groups in the theory of wavelets and introduce a version of Fourier analysis on crystal groups that leads to an explicit description of their group C*-algebras.

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

Remi Gribonval (INRIA Rennes)

Sparse dictionary learning in the presence of noise and outliers
Dienstag, den 3.12.2013, 16.15 Uhr in MA 313
Abstract:

A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.

This is joint work with Rodolphe Jenatton & Francis Bach.

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

Amit Singer (Princeton U)

Three-dimensional structure determination of molecules without crystallization
Mittwoch, den 8.01.2014, 18.00 Uhr in MA 005
Abstract:

Cryo-electron microscopy (EM) is used to acquire noisy 2D projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal is to reconstruct the 3D structure of the macromolecule with sufficiently high resolution. We will discuss algorithms for solving the cryo-EM problem and their relation to other branches of mathematics such as tomography, random matrix theory, representation theory, convex optimization and semidefinite programming.

This is a MATHEON colloquium talk.

Kersten Schmidt (TU Berlin)

Local impedance boundary conditions for wave propagation in viscous gases
Dienstag, den 4.2.2014, 16.15 Uhr in MA 313
Abstract:

In this talk we present Helmholtz like equations for the approximation of the time-harmonic wave propagation in gases with small viscosity (inner friction), which are completed with local boundary conditions on rigid walls. First, we show equations for the pressure, for which the boundary conditions relate the normal derivative of the pressure, the Neumann trace, to the pressure itself, the Dirichlet trace. The boundary condition is of Wentzel type due to a second tangential derivative of the pressure. Then, we introduce Helmholtz like equations for the velocity, where the Laplace operator is replaced by grad div, and the local boundary conditions relate the normal velocity component (Dirichlet trace) to its divergence (Neumann trace). We discuss the variational formulations for those symmetric local boundary conditions, on which finite elements approximations are based on.

The presented models are obtained by asymptotic expansions of the linearized compressible Navier-Stokes equations for gases in rest with no-slip boundary condition on rigid walls. This is a singularly perturbed partial differential equation (PDE) leading to boundary layers in the velocity. The well-known Helmholtz equation with Neumann boundary condition is the limit for vanishing viscosity. The incorporation of small viscosities is especially attractive if one is interesting in the "noise" absoprtion. The method of multiscale expansion is a systematic tool for small viscosity to decompose the solution in so called far fields, which are accurate approximations except in a small vicinity of the boundary, and boundary layer correctors. The far field terms can be defined for the pressure and velocity separately even so they cannot be separated the original equations. This enable us to derive approximative models for either the (far field) pressure or the (far field) velocity, for which the solution can be resolved by the finite element method without need of very fine meshes in the vicinity of the boundary. The models are approximative and rigorous error estimates in powers of the viscosity are available.

Vladimir Kostic (U Novi Sad)

On matrix nearness problems: distance to delocalization/localization
Dienstag, den 11.2.2014, 16.15 Uhr in MA 313
Abstract:

Numerous problems in mechanics, mathematical physics, and engineering can be formulated as eigenvalue problems where the focus is, after determining that the eigenvalues are in the specific desirable domain, to detect minimal size of perturbation that will cause the eigenvalues to leave the domain. The most frequent of such domains are connected to stability of dynamical systems: open left half-plane of the complex plane (continuous dynamical systems) and open unit disk (discrete dynamical systems). While the matrix nearness problems connected to stability have lately been extensively investigated, robustness of spectral inclusions by other domains (that arise for example in acoustic field computations) has remained out of focus.

In this talk, we will consider eigenvalue localization sets of Lyapunov-type which generalize notion of instability/stability to more complicated domains (half-planes, circles, ellipses, rings, cassini ovals, lemniscates, strips, cardioids, etc.). Then, we formulate two matrix nearness problems: distance to delocalization and distance to localization, and discuss methods for their solution.

Finally, we present numerical method that computes distance to delocalisation for Lyapunov-type localization domains. The algorithm that we provide is based on pseudospectral properties, implicit determinant approach and uses Newton’s method in order to optimise minimal singular value over an algebraic curve defined by a Hermitian function in the complex plane.

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

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