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Computing and Visualization in Science

, Volume 16, Issue 6, pp 247–258 | Cite as

Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates

  • Steffen BörmEmail author
  • Knut Reimer
Article

Abstract

Many matrices appearing in numerical methods for partial differential equations and integral equations are rank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of rank-structured matrices are \({\mathcal {H}}^2\)-matrices: they can reach the optimal order of complexity, but are still general enough for a large number of practical applications. We consider algorithms for performing algebraic operations with \({\mathcal {H}}^2\)-matrices, i.e., for approximating the matrix product, inverse or factorizations in almost linear complexity. The new approach is based on local low-rank updates that can be performed in linear complexity. These updates can be combined with a recursive procedure to approximate the product of two \({\mathcal {H}}^2\)-matrices, and these products can be used to approximate the matrix inverse and the LR or Cholesky factorization. Numerical experiments indicate that the new algorithm leads to preconditioners that require \({\mathcal {O}}(n)\) units of storage, can be evaluated in \({\mathcal {O}}(n)\) operations, and take \({\mathcal {O}}(n \log n)\) operations to set up.

Notes

Acknowledgments

A major part of this research was funded by the Deutsche Forschungsgemeinschaft in the context of project BO 3289/4-1.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of KielKielGermany

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