A Note on Preconditioning Nonsymmetric Matrices

Ipsen, Ilse C. F.

doi:10.1137/s1064827500377435

Cited by 145 publications

(132 citation statements)

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“…Lemma 2.1 shows that the higher it is, the more strongly the eigenvalues are clustered. Combining Lemma 2.1 with (1.2), the following theorem is given: [38]. By the above discussion, the choice of the optimal parameter s of the preconditioner H s is equal to 1.…”

Section: It Follows Thatmentioning

confidence: 87%

Block triangular preconditioner for static Maxwell equations

Wu¹,

Huang²,

Li³

2011

Comput. Appl. Math.

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Abstract. In this paper, we explore the block triangular preconditioning techniques applied to the iterative solution of the saddle point linear systems arising from the discretized Maxwell equations. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix are strongly clustered. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner.Mathematical subject classification: 65F10.

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Section: It Follows Thatmentioning

confidence: 87%

Block triangular preconditioner for static Maxwell equations

Wu¹,

Huang²,

Li³

2011

Comput. Appl. Math.

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“…Group Inverse for a Class 2 × 2 Block Matrices. The group inverses of block matrices have various applications in singular differential and difference equations, Markov chains, iterative methods; see [1], [3], [6], [7], [8], [10], [12], [13], [16], [17]. We generalized the results of [2] to the Bezout domain in this section of the paper.…”

Section: By Corollary 24 and (Ab)mentioning

confidence: 99%

Group inverses for matrices over a Bezout domain

Cao¹,

Li²

2009

ELA

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Abstract. Suppose R is a Bezout domain. In this paper, some necessary and sufficient conditions for the existence of the group inverse for square matrix over R are given, the conditions for the existence of the group inverse of products of matrices are studied, and the equivalent conditions for reverse order law of group inverse of product of matrices are obtained. Also the existence and the representation of the group inverse for a class 2 × 2 block matrices over R are studied, and some well known relative results are generalized.

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“…Ipsen [38] proposes three preconditioners as extensions of Murphy, Golub and Wathen's work [44] to systems of form (1.2) with a non-zero (2,2) block (D = 0). The first, a direct analogue of (2.7), serves as an inspiration for our work in Chapter 3, along with the work of de Sturler and Liesen [20].…”

Section: The Preconditioners Of Murphy Golub Wathen and Of Ipsenmentioning

confidence: 99%

“…For the D = 0 case, it was proposed by Murphy, Golub and Wathen [44]. For the D = 0 case, it was proposed by Ipsen [38]. Discussion of these preconditioners will follow in Section 2.1.…”

Section: Introductionmentioning

confidence: 99%

Preconditioners for Generalized Saddle-Point Problems

Siefert¹,

Sturler²

2006

SIAM J. Numer. Anal.

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Generalized saddle point problems arise in a number of applications, ranging from optimization and metal deformation to fluid flow and PDE-governed optimal control. We focus our discussion on the most general case, making no assumption of symmetry or definiteness in the matrix or its blocks. As these problems are often large and sparse, preconditioners play a critical role in speeding the convergence of Krylov methods for these problems. We first examine two types of preconditioners for these problems, one block-diagonal and one indefinite, and present analyses of the eigenvalue distributions of the preconditioned matrices. We also investigate the use of approximations for the Schur complement matrix in these preconditioners and develop eigenvalue analysis accordingly. Second, we examine new developments in probing methods, inspired by graph coloring methods for sparse Jacobians, for building approximations to Schur complement matrices. We then present an analysis of these techniques and their accuracy. In addition, we provide a mathematical justification for their use in approximating Schur complements and suggest the use of approximate factorization techniques to decrease the computational cost of applying the inverse of the probed matrix.Finally, we consider the effect of our preconditioners on four applications.Two of these applications come from the realm of fluid flow, one using a finite element discretization and the other using a spectral discretization.The third application involves the stress relaxation of aluminum strips at low stress levels. The final application involves mesh parameterization and flattening.iii For these applications, we present results illustrating the eigenvalue bounds on our preconditioners and demonstrating the theoretical justification of these methods. We also present convergence and timing results, showing the effectiveness of our methods in practice. Specifically the use of probing methods for approximating the Schur compliment matrices in our preconditioners is empirically justified. We also investigate the h-dependence of our preconditioners one model fluid problem, and demonstrate empirically that our methods do not suffer from a deterioration in convergence as the problem size increases.iv AcknowledgmentsThere are many people deserving of thanks for their assistance in making this thesis a reality, but none more so than my advisor, Eric de Sturler.His sage advice, thoughtful insights and all-around good company played no small role in making this work happen. I am deeply grateful to him for everything he has done! I would also like to thank Michael Heath for his role as the Numerical Analysis group "patriarch." As the unofficial advisor to all of the graduate students, Mike's wisdom (and critique, when warranted) has helped me to hone my presentation skills and helped me to succinctly explain my work to my professional colleagues.The other members of my committee, Robert Skeel and Armand Beaudoin, are also deserving of special thanks, Bob for coming all the way b...

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A Note on Preconditioning Nonsymmetric Matrices

Cited by 145 publications

References 2 publications

Block triangular preconditioner for static Maxwell equations

Block triangular preconditioner for static Maxwell equations

Group inverses for matrices over a Bezout domain

Preconditioners for Generalized Saddle-Point Problems

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