Numerik-Oberseminar SS 2004

Potential Theory in the Analysis of Iterative Methods
(Vortragsserie von Prof. Dr. Christopher Beattie, Virginia Tech, Blacksburg, VA, USA),
29.4., 6.5., 3.6., 10.6. und 17.6. jeweils 16:15-18:00 in MA 313

Interessenten sind herzlich eingeladen!

Weitere Vorträge siehe auch:

• Diplomanden- und Doktorandenseminar des Fachgebiets Numerische Mathematik (Gruppen Mehrmann, Nabben, Tischendorf)
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

Rückblick:

• Numerik-Oberseminar WS 2003/04
• Numerik-Oberseminar SS 2003
• Numerik-Oberseminar WS 2002/03
• Numerik-Oberseminar SS 2002
• Numerik-Oberseminar WS 2001/02
• Numerik-Oberseminar SS 2001

Abstracts zu den Vorträgen:

Abstract:
The talk gives an overview of the theory of polar decompositions in spaces equipped with an indefinite inner product, i.e., decompositions of matrices into two factors that are unitary and selfadjoint with respect to an indefinite inner product. The talk is on an elementary level and covers basic properties and applications up to recently obtained results. The main focus are finite dimensional spaces, but also generalizations to Pontryagin and Krein spaces are briefly discussed. This is joint work with Brian Lins, Patrick Meade, André Ran, and Leiba Rodman.

Abstract:
Multigrid and domain decomposition methods are widely used for solving partial differential equations. Strongly connected with domain decomposition methods are the multiplicative and additive Schwarz-type methods. But also the multigrid method is closely related to a Schwarz iteration. In this talk we consider these methods from an algebraic point of view. We present an algebraic theory which gives a number of new convergence results, especially for nonsymmetric problems.

Abstract:
We consider algebraic differential equations with properly stated leading term. The notion of regularity with tractability index 3 of DAE with well mathed coefficients is descussed. Using the methods of numerical linear algebra the index can be calculated practicaly. A relation between tractability and strangeness index, in case tractability index < 3, will be discussed.

Abstract:
The sphere, which is an important setting for geomathematics and many other areas of application, presents significant challenges for approximation methods. In this talk, for a general mathematical audience, we explore problems of interpolation and integration on the sphere, which have as a common theme the problem of choosing "good" point distributions on the sphere.

Abstract:
We consider wave problems with open frontiers of computational domains. A method of obtaining and effective numerical implementation of non-local boundary conditions is discussed. The method is based on direct analytical or numerical representation of desired boundary operators by using spatial Fourier transformation and time convolution. Different examples of wave problems illustrate are considered.

Abstract:
The paper reviews the state of the art in modeling flow, mass and heat transport in porous media, including conceptual models, governing balance equations, constitutive relations, and numerical methods for solving the resulting multifield problems. The discussion of numerical methods is primarily focused on finite elements and addresses strategies for solving the spatio-temporal processes, velocity approximations, upwind strategies, using discrete feature elements, and adaptive techniques. Software aspects will also be addressed. Along the commercial FEFLOW package we discuss the development status and the practical requirements in today’s subsurface simulation software. The need for coupling different models via programming and data interfaces is emphasized, which allows for example groundwater-river system interactions, parameter estimation or problem optimization in a better and more efficient way. We give examples of field-related applications to illustrate specific challenges of further developments.

Abstract:
The talk is subdivided into three parts. In the first part I give a brief and incomplete overview of the existing methods for the solution of large scale matrix equations, namely Lyapunov, Sylvester and Riccati equations. In the second part I want to concentrate on specialised methods that take into account global low rank structures in the system which is naturally given in the context of model reduction. In this class multigrid methods seem to be the fastest available but also the least versatile. In the last part we consider local low rank structures and investigate how one can formulate solution methods that take this structure into account. This will lead to algorithms based on hierarchical matrices which are versatile but require much more computational effort.

Abstract:
We are interested in the stabilization of the Navier-Stokes equations around a stationary unstable solution by means of a boundary control. We want to find this control in feedback form, that is as a function of the velocity field. For that we look for a feedback law for a linearized problem, and we prove that this feedback law stabilizes the Navier-Stokes equations. The Navier-Stokes equations (as well as the linearized equations around the stationary solution) with a nonhomogeneous Dirichlet boundary condition (corresponding to the control), can be rewritten in the form of an evolution equation for the projection of the velocity field on the Stokes space, coupled with a quasi-stationary solution for the other part of the velocity field. This leads to a nonstandard Riccati equation, which is different of the one obtained for LQR problems associated with parabolic equations. Numerical tests will be presented for the stabilization of a wake behind a cylinder (in 2D), showing the efficiency of the feedback law.

Abstract:
We consider nonnegative matrices as matrices in the max-algebra, i.e. we replace in all operations the usual addition by the max-operation. After an overview we consider some newer applications, in particular in the context of the analytic hierarchy process.

Abstract:
There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. Such methods usually require less storage or computational effort per iteration than methods using an orthonormal basis optimal methods), but the convergence may be delayed. Truncated and augmented Krylov subspace methods and other examples of non-optimal bases, have been shown to converge in many situations, often with small delay, but not in others. We explore the question on what is the effect of having a non-optimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. (joint work with Valeria Simoncini)

Abstract:
The finite mass method is a Lagrangian method for the numerical simulation of compressible flows that I developed during the last years and that arose from a cooperation with astrophysicists in Tübingen. In contrast to the finite volume and the finite element method, the finite mass method is founded on a discretization of mass, not of space. Mass is subdivided into small mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can undergo arbitrary linear deformations. Second order convergence has been proven for motions in external force and velocity fields and for the acoustic equations which result from a linearization of the Euler equations around a constant state. For the full Euler and Navier-Stokes equations, limits exist which satisfy the basic physical principles underlying these equations and can, in this sense, be regarded as solutions of these equations.

Abstract:
This talk presents the results of my PHD thesis which I wrote for the most part at the Institut für Mechanik, TU Berlin under supervision of Prof. Gerd Brunk and partly at the Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), Prof. Rohan Abeyaratne. The influence of spatial discretization (FEM, FDM, MBS) on wave propagation (in unbounded systems) and eigenfrequencies (in bounded systems) is discussed for selective systems of structural mechanics such as (linear) vibrations of strings and beams and the “peridynamic” bar, a specific non-local theory of continuum mechanics as proposed by S. A. Silling in JMPS, 2000: “Reformulation of elasticity theory for discontinuities and long-range forces”.