A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations

Abstract
We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off-diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner.

Citation
Bağcı, H., Pasciak, J. E., & Sirenko, K. Y. (2014). A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations. Numerical Linear Algebra with Applications, 22(2), 371–392. doi:10.1002/nla.1961

Acknowledgements
This work was supported in part by the National Science Foundation through grant DMS-0609544. It was also supported in part by award number KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).

Publisher
Wiley

Journal
Numerical Linear Algebra with Applications

DOI
10.1002/nla.1961

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