A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems

When solving the symmetric positive definite (SPD) linear system ${\bf A} {\bf x}^\star = {\bf b}$ with the conjugate gradient method, the smallest eigenvalues in the matrix ${\bf A}$ often slow down the convergence. Consequently if the smallest eigenvalues in ${\bf A}$ could somehow be "remove...

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Veröffentlicht in:SIAM journal on scientific computing Jg. 27; H. 5; S. 1760 - 1786
Hauptverfasser: Giraud, Luc, Ruiz, Daniel, Touhami, Ahmed
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
Schlagworte:
Algorithms
Eigenvalues
Exact sciences and technology
Mathematics
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Partial differential equations, boundary value problems
Sciences and techniques of general use
Invariant subspace
Decomposition method
Eigenvalue
Iterative method
Partial differential equation
Direct method
Finite element method
65F50
Efficiency
Multigrid
two-grid schemes
Convergence rate
65F15
65H10
iterative methods; 65F10
Conjugate gradient method
Spectral method
Filtering
Convergence acceleration
Numerical linear algebra
Chebyshev polynomial
Initial value problem
Computational complexity
Factorization
Discretization method
block Lanczos method
Sensitivity
Numerical analysis
Boundary value problem
Linear system
Scientific computation
Matrix inversion
Deflation
spectral preconditioning
Domain decomposition
Preconditioning
65N22
Chebyshev polynomials
ISSN:1064-8275, 1095-7197
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Zusammenfassung:When solving the symmetric positive definite (SPD) linear system ${\bf A} {\bf x}^\star = {\bf b}$ with the conjugate gradient method, the smallest eigenvalues in the matrix ${\bf A}$ often slow down the convergence. Consequently if the smallest eigenvalues in ${\bf A}$ could somehow be "removed," the convergence may be improved. This observation is of importance even when a preconditioner is used, and some extra techniques might be investigated to further improve the convergence rate of the conjugate gradient on the given preconditioned system. Several techniques have been proposed in the literature that consist of either updating the preconditioner or enforcing conjugate gradient to work in the orthogonal complement of an invariant subspace associated with the smallest eigenvalues. The goal of this work is to compare several of these techniques in terms of numerical efficiency. Among various possibilities, we exploit the Partial Spectral Factorization algorithm presented in [M. Arioli and D. Ruiz, Technical Report RAL-TR-2002-021, Rutherford Appleton Laboratory, Atlas Center, Didcot, Oxfordshire, England, 2002] to compute an orthonormal basis of a near-invariant subspace of ${\bf A}$ associated with the smallest eigenvalues. This eigeninformation is used in combination with different solution techniques. In particular we consider the deflated version of conjugategradient. As representative of techniques exploiting the spectral information to update the preconditioner we consider also the approaches that attempt to shift the smallest eigenvalues close to one where most of the eigenvalues of the preconditioned matrix should be located. Finally, we consider an algebraic two-grid scheme inspired by ideas from the multigrid philosophy. In this paper, we describe these various variants and we compare their numerical behavior on a set of model problems from Matrix Market or arising from the discretization via the finite element technique of some two-dimensional (2D) heterogeneous diffusion PDE problems. We discuss their numerical efficiency, computational complexity, and sensitivity to the accuracy of the eigencalculation.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1064-8275
1095-7197
DOI:10.1137/040608301