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 |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2006
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Touhami, Ahmed Giraud, Luc Ruiz, Daniel |
| Author_xml | – sequence: 1 givenname: Luc surname: Giraud fullname: Giraud, Luc – sequence: 2 givenname: Daniel surname: Ruiz fullname: Ruiz, Daniel – sequence: 3 givenname: Ahmed surname: Touhami fullname: Touhami, Ahmed |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems |
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