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Host: Kent State University
Homepage: http://etna.mcs.kent.edu/~conference/
Organizers: Lothar Reichel, Daniela Calvetti
Description:
Analysis, Matrix Theory and Approximation Theory have played a key role in the
development and advancement of scientific computation. In the last two decades, as
scientific computation has become a mature field on its own, owing in part to the great
developments in computer technology, unexpected and exciting avenues have been
opened by clever combinations of tools and results from these different areas of
mathematics. We illustrate this with a few examples:
The analysis of iterative methods for the solution of large linear systems of equations uses results and tools from both approximation theory and matrix theory. This is well illustrated in the classical monograph ``Matrix Iterative Analysis'' by Richard Varga. Results from approximation theory yield rates of convergence of iterative methods both for linear systems with symmetric and nonsymmetric matrices. The development of new iterative methods and preconditioners by combining and extending results from matrix theory and approximation theory continues to be an active area of research. Properties of numerical methods for the solution of parabolic partial differential equations are intimately connected with polynomial and rational approximation of the exponential function. Varga pioneered this line of research. Recent work on the approximation of matrix exponentials using Krylov subspace techniques uses and extends well known results in approximation theory and has given rise to new powerful integration methods. Rational approximation of transfer functions plays an important role in signal processing and control theory. In this application the rational functions often are Pad'e approximants that are required to match a certain number of moments. Varga has made significant contributions to rational approximation and in particular Pad'e approximation. In recent applications the Pad'e approximants are computed by the Lanczos process. Block-versions are applied to the simulation of electrical circuits or compact disk players. The aim of these simulations is to replace a complicated system by a simpler one. The solution of two-point boundary value problems can be expressed in terms of piecewise polynomials and properties of numerical solution methods, for instance Galerkin or collocation methods, can be investigated by using results and tools from approximation theory. Varga and collaborators have written several important papers in this area. Generalization to several space dimensions continues to be an active area of research. The Alternating Direction Implicit (ADI) iterative method is popular for the solution of problems in computational fluid dynamics and is also used in multigrid methods. Its properties can be understood by investigating certain rational approximation problems with zeros and poles in disjoint compact sets in the complex plane. There are electrostatic analogues of these approximation problems, where the zeros and poles correspond to negative and positive point charges. Asymptotically optimal iteration parameters can be determined by first replacing the discrete charges by certain Borel measures and then discretizing the latter. Recent work removes the requirement of strict alternation between directions. New applications include image restoration. Early work on the ADI iterative method by Varga showed the connection between approximation theory and this iterative method. The investigation of problems and conjectures in approximation theory can be done by using high precision computer arithmetic. Conjectures on the approximation of exp(x) or |x| on certain real intervals by rational functions have been studied by Varga and collaborators by using high precision computer arithmetic. This approach has also been used to study the Riemann zeta-function with the aim of shedding light on the Riemann hypothesis.
The areas listed continue to be active fields of research. Progress requires almost always that tools and results from several disciplines are used. Richard Varga's contributions are so significant because he mastered several of these fields.
A characterizing feature of the proposed conference is the interplay between topics that at first appear so distant, yet are so intimately related. The expertise of the speakers and the background of the participants will range from analysis, matrix theory and approximation theory to scientific computing. A goal of this meeting is to spur interaction and collaboration between participants with different expertise.
Date received: March 04, 1999
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