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

• Published in: arXiv.org
• Main Authors: Börm, Steffen, Reimer, Knut
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.07.2014
• Subjects: Algorithms; Cholesky factorization; Complexity; Integral equations; Mathematical analysis; Mathematics - Numerical Analysis; Matrix methods; Numerical methods; Partial differential equations; Structured matrices
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1402.5056

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 method 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.