Kolloquium der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen SS 2006


Verantwortliche Dozenten: Alle Professoren der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen
Koordination: Christian Mehl
Termine:Di 16-18 in MA 313 oder n.V.
Inhalt: Vorträge von Gästen und Mitarbeitern- zu aktuellen Forschungsthemen

Vollständige Terminplanung:
 
Datum Zeit Raum Vortragende(r) Titel
Di 25.4.2006 16:15 MA 313 Dr. Olaf Weckner
(The Boeing Math Group)
The Peridynamic theory, its numerical implementation in the EMU code and its applications at The Boeing Company (Abstract)
Di 9.5.2006 16:15 MA 313 Prof. Dr. Martin Gander
(Uni Genf, Schweiz)
Optimal and Optimized Schwarz Methods (Abstract)
Di 23.5.2006 16:15 MA 313 Prof. Dr. Daniel Szyld
(Temple U, Philadelphia, PA, USA)
Optimal Left and Right Additive Schwarz Preconditioning of Minimal Residual Methods with Euclidean and Energy Norms (Abstract)
Di 13.6.2006 16:15 MA 313 Dr. Jochen Garcke
(Canberra, Australien)
Data Mining with Sparse Grids (Abstract)
Do 22.6.2006 14:15 MA 850 Dr. Boris Vexler
(Linz, Österreich)
Optimal Dirichlet Boundary Control of Elliptic and Parabolic Equations (Abstract)
Di 27.6.2006 16:15 MA 313 Prof. Dr. Dieter Hänel
(Uni Duisburg-Essen)
Entwicklungen und Anwendungen von Lattice-Boltzmann-Methoden zur Strömungssimulation (Abstract)
Di 11.7.2006 16:15 MA 313 Prof. Dr. Michael Růžička
(Universität Freiburg)
Numerical Analysis of Generalized Newtonian Fluids (Abstract)
Do 13.7.2006 16:15 MA 313 Prof. Dr. Moshe Goldberg
(Technion, Haifa, Israel)
Minimal Polynomials and Radii of Elements in Finite-Dimensional Power-Associative Algebras (Abstract)
 

Interessenten sind herzlich eingeladen!


Weitere Vorträge siehe auch:

Rückblick:

Rückblick auf das Numerik-Oberseminar, den Vorläufer dieses Kolloquiums:


Abstracts zu den Vorträgen:


Dr. Olaf Weckner, (The Boeing Math Group)
The Peridynamic theory, its numerical implementation in the EMU code and its applications at The Boeing Company
Di 25.4.2006, 16:15 Uhr in MA 313

Abstract:
In this talk I intend to give an introduction to the Peridynamic theory of continuum mechanics (see Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175-209). Using Fouriertransforms the governing integro-differential equation (IDE) can be solved analytically for the simplified linear infinite 1D case. This provides important insight into the fundamental differences between this formulation and the classical formulation of continuum mechanics which relies on partial differential equations (PDE). In particular the propagation of elastic waves and the emergence of discontinuities are compared. The analytic solutions can also be used to evaluate different numerical approaches for the spatial discretization of the IDE such as Gauss-Hermite quadrature, composite midpoint rule and FEM. These numerical studies were the starting point for a present research project between the TU Berlin (lead by Dr. E. Emmrich, Institute of Mathematics) and The Boeing Company. The scope of the project is to improve the spatial quadrature presently used in the EMU code. Finally I present some simulation results involving failure of composite materials which are of current interest at Boeing.


Prof. Dr. Martin Gander, (University of Geneva, Schweiz)
Optimal and Optimized Schwarz Methods
Di 9.5.2006, 16:15 Uhr in MA 313

Abstract:
Optimal and optimized Schwarz methods are based on the classical Schwarz algorithm, but they use instead of Dirichlet transmission conditions more general transmission conditions between subdomains, with the purpose to enhance the convergence speed, to permit methods to be used without overlap, and to obtain convergent methods for problems for which the classical Schwarz method is not convergent, like for example for Helmholtz problems.

Optimal Schwarz methods use integral operators at the interfaces, and converge in a finite number of steps for certain geometric configurations, but at a significantly higher computational cost per iteration compared to the classical Schwarz method. Optimized Schwarz methods use local transmission conditions and converge much faster than classical Schwarz methods, at the same cost per iteration.

I will give an introduction to optimal and optimized Schwarz methods at the algebraic level, and I will show for model problems what changes at the algebraic level are needed to obtain significantly faster methods. I will conclude with an outlook on current research in optimized Schwarz methods.


Prof. Dr. Daniel Szyld, (Temple University, Philadelphia, USA)
Optimal Left and Right Additive Schwarz Preconditioning of Minimal Residual Methods with Euclidean and Energy Norms
Di 23.5.2006, 16:15 Uhr in MA 313

Abstract:
For the solution of non-symmetric or indefinite linear systems arising from discretizations of elliptic problems, two-level additive Schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem exist which are independent of the mesh and the number of subdomains. These bounds are based on some kind of {\em energy norm}. However, in practice iterative methods which minimize the Euclidean norm of the residual are used, despite the fact that the usual bounds are non-optimal, i.e., the quantities appearing in the bounds may depend on the mesh size; see [X.-C.\ Cai and J.\ Zou, {\em Numer.\ Linear Algebra Appl.}, 9:379--397, 2002]. In this paper, iterative methods are presented which minimize the same energy norm in which the optimal Schwarz bounds are derived, thus maintaining the Schwarz optimality. As a consequence, bounds for the Euclidean norm minimization are also derived, thus providing a theoretical justification for the practical use of Euclidean norm minimization methods preconditioned with additive Schwarz. Both left and right preconditioners are considered, and relations between them are derived. Numerical experiments illustrate the theoretical developments. (joint work with Marcus Sarkis)


Dr. Jochen Garcke, (Canberra, Australien)
Data Mining with Sparse Grids
Di 13.6.2006, 16:15 Uhr in MA 313

Abstract:
After a short introduction into the tasks arising in data mining the predictive modelling tasks of regression and classification are considered in more detail.
I will present a generalisation of the sparse grid combination technique for regression in moderately high dimensions d < 15. Here, a regularised least squares approach is discretised and solved on a certain sequence of anisotropic grids. The sparse grid solution is then obtained from the (partial) solutions on these different grids by their linear combination.
In contrast to the original combination technique the coefficents in the combination formula do not depend only on the used partial grids, but on the function to be learned as well, i.e., on the given data. The coefficients are computed to fulfill a certain optimality condition in a projection sense. With this modified ansatz one is able to address instability issues of the normal combination technique which arise in regression applications.
We will present results for a range of benchmark data sets for regression showing the feasibility of this new ansatz in comparison to the normal combination technique as well other standard algorithms.


Dr. Boris Vexler, (Linz, Österreich)
Optimal Dirichlet Boundary Control of Elliptic and Parabolic Equations
Do 22.6.2006, 14:15 Uhr in MA 850

Abstract:
We consider different formulations of optimal Dirichlet boundary control problems with control constraints. Our aim is to compare these formulations with respect to convergence properties of optimization algorithms and of discretization schemes. We present numerical examples illustrating our results.


Prof. Dr. Dieter Hänel, (Universität Duisburg-Essen)
Entwicklungen und Anwendungen von Lattice-Boltzmann-Methoden zur Strömungssimulation
Di 27.6.2006, 16:15 Uhr in MA 313

Abstract:
In diesem Beitrag wird zunächst ein kurzer überblick über die Theorie der Lattice-Boltzmann (LB-)Methoden gegeben, um dann auf einige algorihtmische Weiterentwicklungen, wie genauere Randformulierungen oder lokale Gitterverfeinerungen einzugehen. LB-Methoden, z.T. in Kombination mit Finite-Differenzen-Verfahren,  bieten mittlerweile einen breiten Anwendungsbereich. Beispielhaft aus unseren Arbeiten sind Lösungen der Navier-Stokes-Gleichungen für chemisch reagierende Strömungen, Strömungen mit  Partikelphase, in Atemwegen oder aerodynamische Strömungen.


Prof. Dr. Michael Růžička, (Universität Freiburg)
Numerical Analysis of Generalized Newtonian Fluids
Di 11.7.2006, 16:15 Uhr in MA 313

Abstract:
The numerical analysis of problems with p-structure is quite different from problems which are linear in the main elliptic term. E.g., in order to get optimal error estimates one needs to measure the error not in the usual W^{1,p}-norm, but in a so-called quasi-norm or some equivalent quantity. In the talk we present some recent progress which provides optimal results for different problems with p-structure.


Impressum Christian Mehl 6.9.2006