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Computing and Visualization in Science

, Volume 17, Issue 3, pp 135–150 | Cite as

\(\mathcal H\)-FAINV: hierarchically factored approximate inverse preconditioners

  • R. Kriemann
  • S. Le BorneEmail author
S.I.: EMG 2014

Abstract

Given a sparse matrix, its LU-factors, inverse and inverse factors typically suffer from substantial fill-in, leading to non-optimal complexities in their computation as well as their storage. In the past, several computationally efficient methods have been developed to compute approximations to these otherwise rather dense matrices. Many of these approaches are based on approximations through sparse matrices, leading to well-known ILU, sparse approximate inverse or factored sparse approximate inverse techniques and their variants. A different approximation approach is based on blockwise low rank approximations and is realized, for example, through hierarchical (\(\mathcal H\)-) matrices. While \(\mathcal H\)-inverses and \(\mathcal H\)-LU factors have been discussed in the literature, this paper will consider the construction of an approximation of the factored inverse through \(\mathcal H\)-matrices (\(\mathcal H\)-FAINV). We will describe a blockwise approach that permits to replace (exact) matrix arithmetic through approximate efficient \(\mathcal H\)-arithmetic. We conclude with numerical results in which we use approximate factored inverses as preconditioners in the iterative solution of the discretized convection–diffusion problem.

Keywords

Approximate factored inverse Hierarchical matrices Preconditioning 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Hamburg University of TechnologyHamburgGermany

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