Atlas home || Conferences | Abstracts | about Atlas
Host: University of Colorado
Sponsor: American Mathematical Society, National Science Foundation
Homepage: http://www.ams.org/meetings/src-brualdi.html
Email: wsd@ams.org
Organizers: Richard Brualdi (University of Wisconsin, Madison), Gene Golub (Stanford University), Franklin Luk (Rensellaer Polytechnical Institute), Vadim Olshevsky (Georgia State University) (chair)
Deadline for abstracts: February 01, 1999
Description:
Introduction
Many important problems in pure and applied mathematics and engineering can be reduced to linear algebra problems. Unfortunately, practical circumstances impose limitations on the use of available standard linear algebra methods. For example, in many applications the size of the associated matrices is prohibitively large, so the available standard methods often require an extremely large amount of arithmetic operations.
This is one reason why one seeks in various applications to identify special/characteristic structures that may be assumed in order to speed-up computations. Such additional assumptions are often provided by particular physical properties leading to various structured matrices, such as Toeplitz, Hankel, Vandermonde, Cauchy, Pick matrices, Bezoutians, and others. The structure of these dense matrices is understood in the sense that their entries are defined by a smaller number of parameters. So exploiting such structures allows one to obtain nice solutions for many applied problems as well as to design efficient fast algorithms to compute these solutions.
Structured matrices are encountered in a surprising variety of areas and algorithms, including Pade approximations; continuous fractions; classical algorithms of Euclid, Schur, Nevanlinna, Lanzcos, Levinson; and their generalizations and applications.
Two Examples
1. Operator theory. In the classical Nevanlinna-Pick interpolation problem one seeks a rational interpolant whose norm is bounded by unity in the right half-plane. For this problem the well-known Pick solvability condition (1916) and the Nevanlinna algorithm (1919) both involve a certain structured matrix called the Pick matrix.
2. Electrical engineering. In the now classical N. Wiener monograph Extrapolation, Interpolation and Smoothing of Stationary Time Series a linear prediction problem was reduced to recursive, solving the so-called Yule-Walker equations whose coefficient matrix has the Toeplitz structure.
Further Progress
These problems were among those seeds that grew into deep studies of structured matrices in linear algebra, operator theory, numerical analysis, theoretical computer science, and electrical engineering.
Interpolation
There is a vast operator theory literature on far-reaching generalizations of passive interpolation of the Nevanlinna-Pick type; we mention only that deep results were obtained in the frameworks of several "languages", including the band extension method, the Buerling-Lax-theorem approach, the state-space approach, and lifting-of-commutants method.
Electrical Engineering
In the framework of system and circuit theories, interpolants arise as transfer functions, so passivity is naturally imposed by the conservation of energy. Thus, it is not surprising that fruitful connections to many applied areas were discovered. Many applications such as model reduction, sensitivity minimization, and robust stabilization have been addressed in this way.
Matrix Analysis
It turns out that many nice results and especially many important fast algorithms that were initially obtained for specific patterns of structure can be naturally carried over to the more general important classes of matrices having what is now called displacement structure.
Numerical Analysis
In floating point arithmetic, where the roundoff errors are present, the crucial factor that makes an algorithm practical is its numerical accuracy. Unfortunately, many fast algorithms suffer from often catastrophic propogation of roundoff errors, so one can say that they are often efficient ways to compute "garbage solutions". Moreover, these two targets have even been incorrectly regarded as being unattainable simultaneously, thus leading to the folk conjecture "one has to sacrifice accuracy for speed".
It is remarkable that the results of recent years reveal that the above two targets not only do not conflict with each other but in fact do just the opposite: proper and careful use of structure allows one to design more accurate fast algorithms that can be even better than the standard numerically stable algorithms.
Conference Scope and Topics
Though special sessions and minisymposia on structured matrices are usually included in the programs of the ILAS, SIAM, SPIE, and MTNS conferences, their narrow frameworks usually allow us to focus on one specific application only. The purpose of this conference is to foster integration between different areas and to bring together leading researchers working on all aspects of structured matrices.
There will be several invited tutorial lectures. Contributed talks will focus on recent advances in the following areas: fast algorithms for structured matrices, displacement structure, abstract interpolation, computer arithmetic, numerical accuracy, applications of structured matrices in system theory, circuits, signal processing, adaptive filtering, control, image processing, and preconditioning.
Speakers: Dario Bini (University of Pisa), Patrick Dewilde (Delft University), Israel Gohber (Tel Aviv University), Georg Heinig (Kuwait University), Rien Kaashoek (Vrije University, Amsterdam), Tom Kailath (Stanford University), Franklin Luk (Rensselaer Polytechnical Institute), Vadim Olshevsky (Georgia State University), Haesun Park (University of Minnesota), Bob Plemmons (Wake Forest University), Philip Regalia (Institut National des Telecommunications), Lothar Reichel (Kent State University), Leiba Rodman (College of William and Mary)
Mail Address:
Summer Research Conferences Coordinator American Mathematical Society P.O. Box 6887 Providence, RI 02940
Date received: February 18, 1999
© 2008 Atlas Conferences Inc.