Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method

Faber, Vance; Manteuffel, Thomas A.

doi:10.1137/0721026

Cited by 228 publications

(143 citation statements)

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“…If B is compact, then it can be approxim ated by its finite-dimensional restrictions, hence for large s the truncated algorithm will be close to the full one, and the same can be expected for compact perturbations of the identity. The exact formulation of this is not the aim of this paper, since even in the finite-dimensional case the normal degree n of B is generally large unless n = 1 (see [4,7]). Instead, Theorem 1 will be used in the sequel for the case n = 1 under symmetric part preconditioning in order to verify th a t hereby the truncated GCG-LS(0) m ethod coincides with the full version.…”

Section: T H E Full and Tru N Ca Ted Version S O F Th E C G Mmentioning

confidence: 99%

“…An early paper dealing w ith generalized conjugate gradient m ethods for nonsymmetric systems is [3], and a survey of available m ethods can be found in [14], which includes also the popular GMRES m ethod [13]. The im portant issue of autom atic truncation of the algorithm for a general initial residual was settled independently in [7] and [16]. A discussion about the role played by the initial residual on truncation can be found in [2] and [4].…”

Section: Introductionmentioning

confidence: 99%

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Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

Axelsson

Karátson

2003

Numerical Functional Analysis and Optimization

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Dedicated to the memory of Jean-Jacques Lions: a source of inspiration for rigour in Applied M athem atics A b s tr a c t T he conjugate gradient m ethod for non-sym m etric linear operators in H ilbert space is investigated. C onditions on th e coincidence of th e full and tru n cated versions, known from the finite-dim ensional case, are extended to th e H ilbert space setting. T he focus is on preconditioning by th e sym m etric p a rt of the operator, in which case estim ates are given for th e resulting condition num ber. A n im p o rtan t m otivation for this study is given by differential operators, for which the obtained estim ates yield m esh independent conditioning properties of th e full CGM , and are in fact achieved by th e sim pler tru n cated version.

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Section: T H E Full and Tru N Ca Ted Version S O F Th E C G Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

Axelsson

Karátson

2003

Numerical Functional Analysis and Optimization

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“…A 2L-PCG method is guaranteed to converge if P, as given in (1.3), is SPD or can be transformed into an SPD matrix; see, e.g., [5] for more details. This is certainly satisfied for BNN and DEF when M is SPD; see [21].…”

Section: Positive Definiteness Of P Mgmentioning

confidence: 99%

A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation

Tang¹,

MacLachlan²,

Nabben³

et al. 2010

SIAM J. Matrix Anal. & Appl.

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Abstract. It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann-Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the twolevel preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results.

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“…On the one hand, the fundamental theorem of Faber and Manteuffel [6] shows that if (1.1) holds for a matrix A and a polynomial of degree s, then orthogonal Krylov subspace bases for A can be generated by an (s + 2)-term Arnoldi recurrence (this condition is not only sufficient but also necessary; see [6] or [14] for more details). For a unitary matrix A with n distinct eigenvalues, A * = p(A) with the smallest possible degree of p being n − 1.…”

Section: Hence (13)mentioning

confidence: 99%

“…Let us put the degrees d p (A) and d r (A) into the picture of short recurrence Krylov subspace methods that is described in the introduction: On the one hand, if A is normal with respect to an HPD matrix B and d p (A) = s, then B-orthogonal Krylov subspace bases for A can be generated with an (s + 2)-term Arnoldi recurrence [6].…”

Section: Lemma 23 Let a Be A Square Matrix And Let B Be An Hpd Matrmentioning

confidence: 99%

When is the Adjoint of a Matrix a Low Degree Rational Function in the Matrix?

Liesen¹

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. We show that the adjoint A + of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that A + = r(A). We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the number and distribution of the eigenvalues of A, and compare the McMillan degree with the normal degree of A, which is defined as the smallest degree of a polynomial p for which A + = p(A). We show that unless the eigenvalues of A lie on a single circle in the complex plane, the ratio of the normal degree and the McMillan degree of A is bounded by a small constant that depends neither on the number nor on the distribution of the eigenvalues of A. Our analysis is motivated by applications in the area of short recurrence Krylov subspace methods.

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Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method

Cited by 228 publications

References 3 publications

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space

A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation

When is the Adjoint of a Matrix a Low Degree Rational Function in the Matrix?

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