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Krylov Subspace Techniques for ReducedOrder Modeling of Nonlinear Dynamical Systems
 Appl. Numer. Math
, 2002
"... Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Kry ..."
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Cited by 61 (4 self)
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Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reducedorder bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the VolterraWiener representation of the bilinear system. It is shown that the twosided Krylov subspace technique matches significant more number of multimoments than the corresponding oneside technique.
Reducedorder modeling techniques based on Krylov subspaces and their use in circuit simulation
, 1998
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Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 51 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Solution of shifted linear systems by quasiminimal residual iterations
 Numerical Linear Algebra
, 1993
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A QMRbased interiorpoint algorithm for solving linear programs
 Math. Programming
, 1997
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Nachtigal, A new Krylovsubspace method for symmetric indefinite linear systems
 in Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics
, 1994
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Lanczostype solvers for nonsymmetric linear systems of equations
 Acta Numer
, 1997
"... Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article ..."
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Cited by 37 (11 self)
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Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by lookahead are also discussed. www.DownloadPaper.ir
Nachtigal, Software for simplified Lanczos and QMR algorithms
 Appl. Numer. Math
, 1994
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A class of spectral twolevel preconditioners
 in SISC
, 2003
"... When solving the linear system Ax = b with a Krylov method, the smallest eigenvalues of the matrix A often slow down the convergence. In the SPD case, this is clearly highlighted by the bound on the rate of convergence of the Conjugate Gradient method (CG) given by e (k) √ κ(A) − 1 A ≤ ( √) κ ..."
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Cited by 19 (8 self)
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When solving the linear system Ax = b with a Krylov method, the smallest eigenvalues of the matrix A often slow down the convergence. In the SPD case, this is clearly highlighted by the bound on the rate of convergence of the Conjugate Gradient method (CG) given by e (k) √ κ(A) − 1 A ≤ ( √) κ(A) + 1 k e (0) A, (1) where e (k) = x ∗ − x (k) denotes the forward error associated with the iterate at step k and κ(A) = λmax denotes the condition number. From this bound it can be said that enlarging λmin the smallest eigenvalues would improve the convergence rate of CG. Consequently if the smallest eigenvalues of A could be somehow “removed ” the convergence of CG will be improved. Similarly for unsymmetric systems arguments exist to explain the bad effect of the smallest eigenvalues on the rate of convergence of the unsymmetric Krylov solver [1, 3, 5]. To cure this, several techniques have been proposed in the last few years, mainly to improve the convergence of GMRES. In [5], it is proposed to add a basis of the invariant space associated with the smallest eigenvalues to the Krylov basis generated by GMRES. Another approach based on a low rank update of the preconditionner for GMRES was proposed by [1, 3]. They consider the orthogonal complement of the invariant subspace associated with the smallest eigenvalues to build a low rank update of the preconditioned system. Finally, in [4] a preconditioner for GMRES based on a sequence of rankone updates is proposed that involves the left and right smallest eigenvectors. In our work, we consider an explicit eigencomputation which makes the preconditioner independent of the Krylov solver used in the actual solution of the linear system. We first present our techniques for unsymmetric linear systems and then derive a variant for symmetric and SPD matrices. We consider the solution of the linear system Ax = b, (2) where A is a n × n unsymmetric non singular matrix, x and b are vectors of size n. The linear system is solved using a preconditioned Krylov solver and we denote by M1 the left preconditioner, meaning that we solve