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Simple non-extensive sparsification of the hierarchical matrices

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Abstract

In this paper, we consider the matrices approximated in \({\mathscr{H}}^{2}\) format. The direct solution, as well as the preconditioning of systems with such matrices, is a challenging problem. We propose a non-extensive sparse factorization of the \({\mathscr{H}}^{2}\) matrix that allows to substitute direct \({\mathscr{H}}^{2}\) solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the \({\mathscr{H}}^{2}\) matrix. In the numerical experiments, we show the consistency of this approach in comparison with the other approximate block low-rank hierarchical solvers, such as HODLR [3], H2Lib [5], and IFMM [11].

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Funding

Section 2 is supported by Russian Foundation for Basic Research grant 17-01-00854, Sections 3 and 4 are supported by Russian Foundation for Basic Research grant 16-31-60095, Sectiom 5 is supported by Russian Science Foundation grant 15-11-00033.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences New York University, 251 Mercer Street, New York, NY, 10012-1185, USA

    Daria A. Sushnikova

  2. Skolkovo Institute of Science and Technology, Nobel St. 3, Skolkovo Innovation Center, Moscow, Moscow Region, 143025, Russia

    Ivan V. Oseledets

  3. Institute of Numerical Mathematics Russian Academy of Sciences, Gubkina St. 8, Moscow, 119333, Russia

    Ivan V. Oseledets

Authors
  1. Daria A. Sushnikova
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  2. Ivan V. Oseledets
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Correspondence to Daria A. Sushnikova.

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Communicated by: Leslie Greengard

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Sushnikova, D.A., Oseledets, I.V. Simple non-extensive sparsification of the hierarchical matrices. Adv Comput Math 46, 52 (2020). https://doi.org/10.1007/s10444-020-09794-y

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  • DOI: https://doi.org/10.1007/s10444-020-09794-y

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