A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization

We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one t...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 34; no. 5; pp. A2507 - A2532
Main Authors: Ho, Kenneth L., Greengard, Leslie
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects:
Algorithms
Applied mathematics
Approximation
Decomposition
Integral equations
Iterative methods
Methods
Sparsity
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson--Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or low-frequency Helmholtz equations, both phases typically have complexity $\mathcal{O} (N)$ in two dimensions, where $N$ is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are $\mathcal{O} (N^{3/2})$ and $\mathcal{O} (N \log N)$ for precomputation and solution, respectively. Extensive numerical experiments show a speedup of ${\sim}100$ for the solution phase over modern fast multipole methods; however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with ill-conditioned linear systems or when the same system is to be solved with multiple right-hand sides. Our algorithm is implemented in Fortran and freely available. [PUBLICATION ABSTRACT]
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ISSN:1064-8275
1095-7197
DOI:10.1137/120866683