Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 39; no. 3; pp. 340 - 370
Main Authors: Tang, J. M., Nabben, R., Vuik, C., Erlangga, Y. A.
Format: Journal Article
Language:English
Published: Boston Springer US 01.06.2009
Springer Nature B.V
Subjects:
Accounting
Algorithms
Coefficients
Computational Mathematics and Numerical Analysis
Convergence
Decomposition
Domain decomposition
Eigenvalues
Equivalence
Linear systems
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Multigrid methods
Robustness
Roundoff error
Spectra
Theoretical
Two-grid schemes
Two-level PCG methods
Multigrid
Two-level preconditioning
SPD matrices
Deflation
Conjugate gradients
Domain decomposition
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-009-9272-6