An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling

Howle, Victoria E.; Vavasis, Stephen A.

doi:10.1137/s0895479800370871

Cited by 44 publications

(18 citation statements)

References 17 publications

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“…Besides special solvers developed with a particular application in mind, like, e.g. those studied in Adams (2007), Elman et al (2001), Erlangga et al (2004), Haber & Ascher (2001), Howle & Vavasis (2005), Poirier (2000) and Reitzinger et al (2003), there has been some work on the use of generalpurpose techniques such as SSOR, polynomial preconditioning, incomplete factorizations and sparse approximate inverses (see, e.g. Freund, 1990;Horesh et al, 2006;Mazzia & Pini, 2003;Mazzia & McCoy, 1999).…”

Section: Previous Workmentioning

confidence: 99%

“…Note that these conditions rule out the Hermitian and skew-Hermitian cases. Complex symmetric systems arise in a number of applications, including wave propagation (Sommerfeld, 1949), diffuse optical tomography (Arridge, 1999, Section 3.3), quantum mechanics (numerical integration of the time-dependent Schrödinger equation by implicit methods) (van Dijk & Toyama, 2007), electromagnetism (Maxwell's equations) (Hiptmair, 2002), molecular scattering (Poirier, 2000), structural dynamics (frequency response analysis of mechanical systems) (Feriani et al, 2000), electrical power system modelling (Howle & Vavasis, 2005) and lattice quantum chromodynamics (Frommer et al, 2000). Another important source of complex symmetric linear systems is the discretization of certain self-adjoint integrodifferential equations arising in environmental modelling (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…on the main diagonal (see, e.g. Bertaccini, 2004;Haber & Ascher, 2001;Howle & Vavasis, 2005;Poirier, 2000). Using complex arithmetic throughout the code would be wasteful.…”

Section: Introductionmentioning

confidence: 99%

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Block preconditioning of real-valued iterative algorithms for complex linear systems

Benzi

Bertaccini

2007

IMA Journal of Numerical Analysis

155

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We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditioners for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation. Numerical experiments illustrating the performance of the various approaches are presented.

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Section: Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Block preconditioning of real-valued iterative algorithms for complex linear systems

Benzi

Bertaccini

2007

IMA Journal of Numerical Analysis

155

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“…Guattery used support-graph theory to bound the condition number of incomplete factorizations without diagonal modification [7]. Howle and Vavasis generalized the preconditioners of Gremban and Miller to complex systems [9]. We are also aware that John Reif of Duke University is working on this subject, but to the best of our knowledge he has not yet published his results.…”

Section: Generalizing the One-dimensional Preconditionermentioning

confidence: 99%

Support-Graph Preconditioners

Bern¹,

Gilbert²,

Hendrickson³

et al. 2006

SIAM J. Matrix Anal. & Appl.

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Abstract. We present a little-known preconditioning technique, called support-graph preconditioning, and use it to analyze two classes of preconditioners. The technique was first described in a talk by Pravin Vaidya, who did not formally publish his results. Vaidya used the technique to devise and analyze a class of novel preconditioners. The technique was later extended by Gremban and Miller, who used it in the development and analysis of yet another class of new preconditioners. This paper extends the technique further and uses it to analyze two classes of existing preconditioners: modified incomplete-Cholesky and multilevel diagonal scaling. The paper also contains a presentation of Vaidya's preconditioners, which was previously missing from the literature. [8]) and multilevel-diagonal-scaling (MDS) preconditioners (see [10], for example).The principle goal of this paper is to bring these techniques to the attention of a wider community of researchers. By doing so, we hope to encourage further work in this promising area. The primary original content of this paper, analyzing known preconditioners using the support-graph technique, serves several purposes. First, the analysis of MICC preconditioners establishes bounds that have never been proved before. Second, it shows that the technique is more widely applicable than previously appreciated. Third, we feel that the new proofs provide useful insights into these preconditioners; these insights can be used to improve the preconditioners and to guide heuristics for the construction of additional preconditioners.A secondary goal of this paper is to provide a complete presentation of the support-graph technique and of Vaidya's preconditioners. Vaidya's important contribution has never been published. Although most of the theory that he uses is presented in Gremban's PhD thesis [6], Vaidya's preconditioners have not been described in any published form. We seek to rectify this situation. Our complete presentation of the support-graph technique is necessary since some important portions of the

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“…Complex symmetric matrices also appear in the consideration of quantum reaction dynamics [2], the numerical simulation of high-voltage insulators [19], and thermoelastic wave propagation [22]. They have also been the focus of recent algorithmic work (see [1,12,14], for example). The basic theory of complex symmetric matrices is discussed in the texts [7,13].…”

Section: 4])mentioning

confidence: 99%

Approximate antilinear eigenvalue problems and related inequalities

Garcia

2007

Proc. Amer. Math. Soc.

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Abstract. If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ(|T |) of √ T * T can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator T : H → H), in terms of the spectra of the selfadjoint operators Re T and Im T , restricting the possible location of elements of σ(|T |). A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.

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An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling

Cited by 44 publications

References 17 publications

Block preconditioning of real-valued iterative algorithms for complex linear systems

Block preconditioning of real-valued iterative algorithms for complex linear systems

Support-Graph Preconditioners

Approximate antilinear eigenvalue problems and related inequalities

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