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Dynamical systems are the principal tool used in modeling a
number of physical phenomena such as weather or behavior of large-scale
integrated circuits. During the design phase of a modern computer processor,
designers must consider as many as 1,000,000 variables to model potential
interference and undesirable electromagnetic effects among the different
circuits within the chip. The overly large number of variables (i.e., the Reduction of the complexity of the model should be done so that -
approximation error is small and measurable, -
essential properties (e.g., stability) of the original system are preserved, and -
the algorithm for reducing the model is computationally stable and efficient.
Singular-value decomposition (SVD) methods are fairly complex computationally and so are generally applicable to models with fairly small complexity (See [1] for a nice overview.); however, SVD methods provide an explicit value for the approximation error. Krylov-based reduction methods require only matrix multiplication and so they are perfectly suitable for models with very large complexity. The disadvantage with Krylov-based approximations is that they lack an error bound, and stability of the original model is not always preserved by the approximating system. The purpose of The text begins with motivating examples before it presents the essential tools from matrix theory and linear dynamical systems along with the Sylvester and Lyapunov equations, which are necessary in the Hankel-norm approximation methods. In this exhaustive, but wonderfully readable, presentation (about 200 pages), the author gives extensive references and citations essential to those new to the field and essential for a preservation of the development of the field. The remainder of the text addresses balanced truncation methods and methods for balancing (e.g., Lyapunov, stochastic, bounded real, and positive real balancing); Hankel-norm approximation, various methods for constructing approximating models, and error bounds; various Krylov-based methods for constructing approximating models; and finally, the purpose of the text, SVD-Krylov methods. The final chapter of the book is a presentation of case studies of the SVD-Krylov approximation techniques.
Finally, one cannot omit from any review of this book a
discussion of its excellent readability. The quality of Antoulas's writing
is comparable to that found in Fritz John's classic text
I. Introduction -
Introduction -
Motivating examples
II. Preliminaries -
Tools from matrix theory -
Linear dynamical systems: Part 1 -
Linear dynamical systems: Part 2 -
Sylvester and Lyapunov equations
III. SVD-based approximation methods -
Balancing and balanced approximations -
Hankel-norm approximations -
Special topics in SVD-based approximation
IV. Krylov-based approximation methods -
Eigenvalue computations -
Model reduction using Krylov methods
V. SVD-Krylov methods and case studies -
SVD-Krylov methods -
Case studies -
Epilogue -
Problems
[1] C. Beattie and S. Gugercin, Approximation and control of large-scale dynamical systems. Found at http://www.modelreduction.math.vt.edu. |

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