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Date Time Venue Talk
11/09/10 02:00 pm Schwarzenbergstrasse 95, Room 3.053 Spline-Ausgleich für die glatte Approximation von NC-Daten (Bachelorarbeitsvortrag)
Michael Seeck

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10/20/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 Entwicklung eines Algorithmus zur effektiven Lösung großer nichtlinearer Gleichungssysteme
Fabian Krome

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09/22/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 Eine graphische Benutzeroberfläche bei Kurvenverfolgung (Studienarbeitsvortrag)
Uwe Köcher

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09/22/10 02:00 pm Schwarzenbergstrasse 95, Room 3.053 Inexakte Inverse Iteration (Diplomarbeitsvortrag)
Fatih Berber

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09/15/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 Aspects of eigenvalue computations using Induced Dimension Reduction (Bachelorarbeitsvortrag)
Olaf Rendel

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06/16/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 Inducing dimension reduction for efficientlysolving large linear systems of equations
Gerard L.G. Sleijpen, Department of Mathematics, Utrecht University, Utrecht, The Netherlands

The Induced Dimension Reduction method was proposed in 1980 by Peter Sonneveld as an iterative method for solving large non-symmetric linear systems of equations. IDR can be considered as the predecessor of methods like CGS (Conjugate Gradient Squared [Sonneveld '89]) and Bi-CGSTAB (Bi-Conjugate Gradients STABilized [van der Vorst '92]). All three methods are based on efficient short recurrences. An important similarity between the methods is that they use orthogonalization with respect to a fixed `shadow residual'. Of the three methods, Bi-CGSTAB has gained the most popularity, and is probably still the most widely used short recurrence method for solving non-symmetric systems.

Recently, Sonneveld and van Gijzen revived the interest for IDR. In 2008, they demonstrate that a higher dimensional shadow space, defined by an n by s matrix tR_0, can easily be incorporated into IDR, yielding a highly effective method. Convergence (in terms of steps, or, equivalently, in terms of matrix-vector multiplications) is often comparable to GRMES, but in contrast to GMRES, this ''s version'' of IDR relies on short recurrences and all steps are equally fast.

The original IDR method is closely related to Bi-CGSTAB. It is therefore natural to ask whether Bi-CGSTAB can be extended to an ''s-version'' in a way similar to IDR. To answer this question we explore the relation between IDR and Bi-CGSTAB. Our findings lead to an abstract description of the IDR method. It shows that there is a lot of freedom in implementing , leading to variants that are mathematically equivalent. The implementational variants, however, may have different stability and efficiency properties.

Bi-CGSTAB relies on degree 1 stabilization polynomials. Higher degree stabilization polynomials can also be exploited as is shown by Sleijpen and Fokkema in 1993. The resulting method BiCGstab(L) is often more stable than Bi-CGSTAB leading the much faster convergence. As shown by Sleijpen, van Gijzen 2009 and Tanio, Sugihara 2009, higher degree stabilization polynomials can also be incorporated in IDR and it can greatly improve stability of IDR with degree 1 stabilization polynomials. We argue that this is another implementational variant of IDR.

This is joint work with Martin van Gijzen, Delft University of Technology, Delft, The Netherlands

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04/14/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 Inverse Iteration, Newton-Abschätzungen und Anwendung auf Rayleigh-Quotienten-Iterationen bei nichtlinearen Eigenwertproblemen
Prof. Hubert Schwetlick, TU Dresden, Institut für Numerische Mathematik

Bekanntlich liefert ein Schriitt $(u,\theta) \mapsto u_+^{InvIt}$ der Inversen Iteration für das nichtlineare Eigenwertproblem $T(\lambda)x=0$ dieselbe Richtung wie ein Schritt $(u,\theta) \mapsto (u_+^{Newt},\theta_+^{Newt})$ des Newtonverfahrens für das erweiterte System $T(\lambda)x=0,\;w^Hx=1$ mit einem geeigneten Skalierungsvektor $w$, d.h., es gilt $\mbox{span}\,\{u_+^{InvIt}\}=\mbox{span}\,\{u_+^{Newt}\}$. Es liegt daher nahe, zur Abschätzung der Verbesserung der Eigenvektorapproximation $u$ durch die Inverse Iteration Newton-Techniken zu verwenden. Es wird gezeigt, dass dies zu genauen Abschätzungen führt, wenn explizit mit dem Restglied zweiter Ordnung gearbeitet und dessen spezielle Produktstruktur berücksichtigt wird wie das von \textsc{Heinz Unger} [50] erstmalig (und ohne publizierten Beweis) für das lineare Problem $T(\lambda)=A-\lambda I$ getan worden ist.

Durch Kombination mit neuen Abschätzunegn für das nichtlineare klassische bzw. verallgemeinerte Rayleigh-Funktional läßt sich dann einfach die quadratische Konvergenz
der nichtlinearen Rayleigh-Funktional-Iteration wie auch die kubische Konvergenz der nichtlinearen Verallgemeinerung der zweiseitigen Ostrowskischen Rayleigh-Quotienten-Iteration herleiten.

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02/17/10 02:00 pm Schwarzenbergstrasse 95, Room 3.053 wird noch bekannt gegeben
Michael Dudzinski

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02/03/10 01:00 pm Schwarzenbergstrasse 95, Room 3.053 On the motion of several rigid bodies in an incompressible non-Newtonian fluid*
Prof. Sarka Necasova, Institute of Mathematics of the Academy of Sciences, Praha, Czech Republic

The motion of one or several rigid bodies in a viscous fluid occupying a bounded domain ­$\Omega in R^3$ represents an interesting theoretical problem featuring, among others, possible contacts of two or more solid objects. We consider the motion of several rigid bodies in a non-Newtonian fluid of a power-law type. Our main result establishes the existence of global-in-time solutions of the associated evolutionary system, when collisions of two or more rigid objects do not appear in a finite time unless they were present initially.

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01/27/10 03:00 pm Schwarzenbergstrasse 95, Room 3.053 A self-similar solution for the porous medium equation in a two-component domain*
Prof. Jan Filo, Comenius University, Bratislava, Slovak Republic

We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition
\[
w^\prime =h^\prime \;,\qquad h=\psi(w)\qquad\text{at the point $\,x=0\,$}.
\]
We show that, for a prescribed power-law nonlinearity $\psi$ and using the solution $(w,h)$, a self-similar solution to the porous medium equation in the two-component domain can be constructed.

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* Talk within the Colloquium on Applied Mathematics