Frobenius norm minimization and probing for preconditioning

Abstract

In this paper we introduce a new method for defining preconditioners for the iterative solution of a system of linear equations. By generalizing the class of modified preconditioners (e.g. MILU), the interface probing, and the class of preconditioners related to the Frobenius norm minimization (e.g. FSAI, SPAI) we develop a toolbox for computing preconditioners that are improved relative to a given small probing subspace. Furthermore, by this MSPAI (modified SPAI) probing approach we can improve any given preconditioner with respect to this probing subspace. All the computations are embarrassingly parallel. Additionally, for symmetric linear system we introduce new techniques for symmetrizing preconditioners. Many numerical examples, e.g. from PDE applications such as domain decomposition and Stokes problem, show that these new preconditioners often lead to faster convergence and smaller condition numbers.

Keywords:

• Preconditioning
• Schur complements
• Probing
• Sparse approximate inverses
• Modified incomplete factorization

Acknowledgements

The first author is grateful to Peter Jimack for contributing a lot of suggestions and motivation during a sabbatical in Leeds.

Additional information

Notes on contributors

A. Kallischko

Email: [email protected]