and

Problems arising in applications give the motivation for studying issues of approximation and model reduction. Some of these problems are: systems described by PDEs, systems arising in circuit simulation, weather prediction, components of mechanical structures (e.g. the Space Station).

Model reduction aims at replacing a system of differential or difference equations of high complexity by one of much lower complexity. In so doing, one tries to preserve certain critical properties of the system (e.g. stability) and approximate well important features (e.g. the system response). During the last two decades, a lot of progress has been made in the theory of this approximation problem. The first part of the course will review the foundations of this theory and will present the key results of frequency and time domain approximations (Grammian based balanced truncation and Hankel norm approximation). More recently, the need has arisen to apply these methods to problems of very high complexity; in such cases the resulting computational complexity becomes prohibitively high and different approaches to the problem have to be developed. In the second part of the course we will present techniques that can be applied to large scale systems provided the models are sparse or structured (Padé like approximations and Krylov based methods).

Attendees will get an up-to-date account of this area with discussion of various application examples. The course should help them apply such ideas to their own area of research.

30% introductory; 30% intermediate; 40% advanced

Basic knowledge of systems theory (state-space models). Some background in linear algebra and numerical linear algebra.

Those interested in a state-of-the-art account of system approximation and model reduction of dynamical systems.

**Thanos Antoulas** obtained his PhD in mathematics at the ETH Zurich in 1980, and since 1982 he has been with the Department of Electrical and Computer Engineering, Rice University. He was awarded Fellowships of the IEEE (Institute of Electrical and Electronics Engineers.) and the JSPS (Japan Society for the Promotion of Science). For the past 5 years he has been Editor-in-Chief of Systems and Control Letters. His research interests are in the broad area of modeling and control of complex dynamical systems.

**Paul Van Dooren** obtained his PhD in applied mathematics in 1979 at the Katholieke Universiteit Leuven, Belgium. He held positions at the Philips Research Laboratory (80-91), the University of Illinois. Urbana-Champaign (91-94), and the Universite Catholique de Louvain (94-now). He is Editor-in-Chief of the SIAM Journal of Matrix Analysis and Applications. He received the Householder Award and the Wilkinson Prize of Numerical Analysis and Scientific Computing. His interests are in numerical linear algebra and in systems and control theory.

Sunday, July 9, 2000

I. Thanos Antoulas

- Introduction to Approximation of State-Space Models
- Grammians of State-Space Models and Balanced Realizations
- Balanced Truncation and Hankel Norm Approximations
- Approximation by Moment Matching and Rational Interpolation

II. Paul Van Dooren

- Padé Approximations and the Lanzcos Algorithm
- Multi Point Pade Approximations
- Krylov Space Methods

III. Thanos Antoulas and Paul Van Dooren

- Recursive Modeling and the Lanczos Algorithm (TA)
- Extensions to Time Varying Systems (PVD)

IV. Thanos Antoulas and Paul Van Dooren

- Numerical Aspects (PVD)
- Numerical Examples and Case Studies Involving Systems Arising in Several Areas of Application (PVD and TA)

Seats are limited. Please register before the deadline. To register, please submit the Preregistration Form. Submit completed form with registration payment to reach SIAM on or before June 7, 2000. Registration fee includes coffee breaks and lunch on Sunday, July 9.

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Created 4/12/00; Updated 4/12/00