Mathematical Institute  Chair of Appl. Math.  Teaching  SS 2004	Lectures in SS 2004	Mathem. Kolloquium
	Seminars in SS 2004	im SS 2004

	Chair of Applied Mathematics
	(MATHE V)
	Mathematisches Kolloquium im SS 2004

Vortragsankündigungen für das Mathematische Kolloquium

Kolloquiumsvortrag am 15. Juli 2004

Herr Priv.-Doz. Dr. Vincent Heuveline Universität Heidelberg

über das Thema

Über Modellreduktion für zielorientierte a posteriori Fehlerschätzer bei partiellen Differentialgleichungen: Anwendung für Mikro-Aggregate .

Kolloquiumsvortrag am 08. Juli 2004

Herr Prof. Dr. Roger Fletcher University of Dundee, UK

über das Thema

A New Look at Newton Methods .

Abstract:

The talk examines the Newton-Raphson method for solving well-determined systems of nonlinear equations r(x)=0, when the method is stabilised by using a line search in which the sum of squares function r^Tr is approximately minimized. This topic is important in its own right, and it also has implications for augmented Lagrangian methods in NLP, and for feasibility restoration in filter methods. We look for a method that does not require second derivatives of r to be calculated, and which is applicable to large systems in which the Jacobian matrix A is sparse.

The Newton-Raphson method with this line search can sometimes converge very slowly to a limit point at which A is singular and the gradient of r^Tr is non-zero, clearly an unacceptable outcome. Powell's dog-leg method avoids this difficulty (in theory) but can converge slowly in the vicinity of a local minimizer of r^Tr at which r^Tr > 0 (necessarily det(A)=0 in this case also).

We examine the geometry around such a local minimizer and show a surprising result (to me) about the surface det(A)=0. This enables a step to be determined which is conjugate (in the limit) to the Newton-Raphson step. Use of the step enables fast (but probably not superlinear) convergence to the local minimizer to be obtained.

In a regular situation, when the Newton-Raphson method converges at second order to an exact solution of r(x)=0, the conjugate step is not required. Also some bias towards the steepest descent direction is likely to be needed to prove global convergence, and also to handle situations in which A has multiple zero eigenvalues. Exactly how to accommodate all these features in a single algorithm is the subject of ongoing research.

Kolloquiumsvortrag am 01. Juli 2004

Herr Prof. Dr. Yu-Hong Dai Chinese Academy of Sciences, Peking, China (z.Zt. Universität Bayreuth)

über das Thema

New Algorithms for A Singly Constrained Class of Quadratic Programs subject to Lower and Upper Bounds .

Kurzfassung:

Kolloquiumsvortrag am 13. Mai 2004

Herr Prof. Dr. Fritz Colonius Institut für Mathematik, Universität Augsburg

über das Thema

Fundamental-Halbgruppen für Dynamische Systeme .

Kurzfassung:

Das Limesverhalten von autonomen gewöhnlichen Differentialgleichungen kann man analysieren, indem man das Verhalten unter kleinen zeitabhängigen Störungen betrachtet. Diese können auch als Steuerungen (d.h. Kontrollfunktionen) interpretiert werden. Die Limesmengen der Differentialgleichung werden dann zu Mengen vollständiger Kontrollierbarkeit "aufgeblasen", denen man, in Analogie zur Homotopietheorie, eine fundamentale Halbgruppe zuordnen kann. Deren inverse Limes für verschwindende Störungen liefert eine Invariante für die Ausgangs-Differentialgleichung.

© Robert Baier ()