Title: Towards effective methods for computing matrix
pseudospectra
Speaker: E. Gallopoulos ( email )
Affiliation: Department of Computer Engineering and
Informatics , University of Patras
26500 Patras, Greece
Note: This talk presents results
obtained in joint reseach with C.
Bekas, E. Kokiopoulou, I. Koutis,
A. Sidiropoulos and V. Simoncini.
Abstract The e-
pseudospectrum of a matrix, defined for example as
| Le (A) = {z: z Î
L(A+E) for some E Î Cn x n with || E || £ e} | |
where
L(A) denotes the spectrum of A, is acknowledged to be a
powerful mechanism for investigating the behavior of several (nonnormal)
matrix-dependent algorithms, ranging from iterative methods for large linear
systems to time-stepping algorithms; note the inclusion of specific functions to
that effect in the Test Matrix Toolbox of
MATLAB as well as the Oxford URL http://web.comlab.ox.ac.uk/projects/pseudospectra.
The standard workhorse method for computing pseudospectra is the following: i)
Discretize the region of interest in the complex plane and ii) compute the
minimum singular value of the matrix zI-A at every gridpoint z. This algorithm
is simple, robust, embarassingly parallel but extremely expensive even for
medium sized matrices: Much more expensive than having to compute matrix
characteristics, such as eigenvalues, singular values and condition numbers. In
other words, pseudospectra are powerful but we must find ways for obtaining them
at lesser cost. ``Domain based'' methods for computing pseudospectra attempt to
reduce the number of gridpoints while ``matrix based'' methods attempt to obtain
the singular values faster. In this talk we introduce the concept of
pseudospectra and present recent work designed to
alleviate this computational bottleneck in order to let the pseudospectrum
become a practical tool for engineers and scientists. We describe algorithms that have drastically improved performance while
maintaining a high degree of large grain parallelism. We also consider the effectiveness of these methods in the context of
parallel architectures as well as a MATLAB-based environment for parallel programming using MPI on
small, off-the-shelf parallel systems.
About the speaker Stratis Gallopoulos (Ph.D.
Illinois, B.Sc. Imperial College, London) is Professor of Computer Engineering
and Informatics at the University of Patras. Prior to that he held positions at
the University of Illinois at Urbana-Champaign, the University of California
Santa Barbara and the NASA Goddard Space Flight Center. He participated in the
development and practical application of the Cedar multiprocessor at the Center
for Supercomputing Research and Development (1987-94) and of the first Massively
Parallel Processor built by Goodyear for the NASA Goddard Space Flight Center
(1980-85). His research interests include numerical algorithms and environments
for large-scale scientific computation, parallel processing and Computational
Science & Engineering education. Professor Gallopoulos served on the
editorial boards of the IEEE Computing in Science & Engineering Magazine, he
was Program Committee Chair of the 2001 ACM International Conference
on Supercomputing, he is member of the Steering Committee of that same
conference as well as of the European Research Conference on Advanced
Environments for High Performance Computing and is editor of the International Journal of High-Speed Computing.
He is also member of program committees for several other conferences, most
recently ICPP'02, HERCMA'01 and ISHPC-IV.
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