Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Pade-type or even true Pade approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation property in the Hermitian case

Krylov subspaces associated with higher-order linear dynamical systems

A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed

A biconjugate gradient type algorithm on massively parallel architectures

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine

A sequential semidefinite programming method and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. In particular, a suitable symmetrization procedure needs to be chosen for the linearization of the complementarity condition. The choice of the symmetrization procedure can be shifted in a very natural way to certain linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The resulting sequential semidefinite programming (SSP) method is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class

QMR: A Quasi-Minimal Residual method for non-Hermitian linear systems

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. A novel BCG like approach is presented called the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included

Krylov Subspace Methods for Complex Non-Hermitian Linear Systems

We consider Krylov subspace methods for the solution of large sparse linear systems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such as inverse scattering, numerical solution of time-dependent Schrodinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically, the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. In this paper, we first describe a Krylov subspace approach with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, we study special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. We also include some results concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported

Quasi-kernel polynomials and convergence results for quasi-minimal residual iterations

Recently, Freund and Nachtigal have proposed a novel polynominal-based iteration, the quasi-minimal residual algorithm (QMR), for solving general nonsingular non-Hermitian linear systems. Motivated by the QMR method, we have introduced the general concept of quasi-kernel polynomials, and we have shown that the QMR algorithm is based on a particular instance of quasi-kernel polynomials. In this paper, we continue our study of quasi-kernel polynomials. In particular, we derive bounds for the norms of quasi-kernel polynomials. These results are then applied to obtain convergence theorems both for the QMR method and for a transpose-free variant of QMR, the TFQMR algorithm

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

AbstractIn recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices have appeared in the literature. In this paper, we study restricted congruence transformations of Hermitian matrices M that, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to the concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain new and shorter proofs for several known inertia theorems and also derive some new results of this type. For some applications, a special class of “almost” unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form, which, in particular, displays the eigenvalues of A. Moreover, this quasi-spectral decomposition is used to derive a generalized signature formula and to study Hermitian matrix pencils