CMOR Technical Reports
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Computational Applied Mathematics and Operations Research (CMOR) technical reports 1981-present. Use the "Browse" bar to filter by date, author, etc.
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Browsing CMOR Technical Reports by Author "Antoulas, A.C."
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Item A Modified Low-Rank Smith Method for Large-Scale Lyapunov Equations(2001-05) Antoulas, A.C.; Sorensen, D.C.; Gugercin, S.In this note we present a modified cyclic low-rank Smith method to compute low-rank approximations to solutions of Lyapunov equations arising from large-scale dynamical systems. Unlike the original cyclic low-rank Smith method introduced by Penzl in [18], the number of the columns in the approximate solutions does not necessarily increase at each step. The number of columns required by the modified method is usually much lower than the original cyclic low-rank Smith method. The modified method never requires more columns than the original. Upper bounds are established for the errors in the low-rank approximate solutions and also for the errors in the resulting approximate Hankel singular values. Numerical results are given to verify the efficiency and accuracy of the new algorithm.Item A Survey of Model Reduction Methods for Large-Scale Systems(2000-12) Antoulas, A.C.; Sorensen, D.C.; Gugercin, S.An overview of model reduction methods and a comparison of the resulting algorithms are presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better.Item Approximation of large-scale dynamical systems: An Overview(2001-02) Antoulas, A.C.; Sorensen, D.C.In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.Item Gramians of Structured Systems and an Error Bound for Structure-Preserving Model Reduction(2004-09) Sorensen, D.C.; Antoulas, A.C.In this paper a general framework is posed for defining the reachability and controllability gramians of structured linear dynamical systems. The novelty is that a formula for the gramian is given in the frequency domain. This formulation is surprisingly versatile and may be applied in a variety of structured problems. Moreover, this formulation enables a rather straightforward development of apriori error bounds for model reduction in the H2 norm. The bound applies to a reduced model derived from projection onto the dominant eigenspace of the appropriate gramian. The reduced models are structure preserving because they arise as direct reduction of the original system in the reduced basis. A derivation of the bound is presented and verified computationally on a second order system arising from structural analysis.Item Lyapunov, Lanczos, and Inertia(2000-05) Antoulas, A.C.; Sorensen, D.C.We present a new proof of the inertia result associated with Lyapunov equations. Furthermore we present a connection between the Lyapunov equation and the Lanczos process which is closely related to the Schwarz form of a matrix. We provide a method for reducing a general matrix to Schwarz form in a finite number of steps (O(n3)). Hence, we provide a finite method for computing inertia without computing eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.Item On the Decay Rate of Hankel Singular Values and Related Issues(2001-05) Antoulas, A.C.; Sorensen, D.C.; Zhou, Y.This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues like the Hankel singular values, are decaying rapidly. This fact has motivated the development of several algorithms for computing low rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.Item Projection Methods for Balanced Model Reduction(2001-03) Sorensen, D.C.; Antoulas, A.C.The purpose of this paper is to investigate projection methods for the iterative computation of partially balanced reduced order systems. This approach is both completely automatic once an error tolerance is specified and yields an error bound. This is to be contrasted with existing projection methods, namely PVL (Padé via Lanczos) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and appoximation of certain system grammians and in particular on the cross grammian.