Randomized Compression of Rank-Structured Matrices Accelerated with Graph Coloring
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arXiv
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| Format: | Preprint |
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2022
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| author | Levitt, James Martinsson, Per-Gunnar |
| author_facet | Levitt, James Martinsson, Per-Gunnar |
| contents | A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the assumption that algorithms for rapidly applying $A$ and $A^{*}$ to vectors are available. The algorithm analyzes the hierarchical tree that defines the rank structure using graph coloring algorithms to generate a set of random test vectors. The matrix is then applied to the test vectors, and in a final step the matrix itself is reconstructed by the observed input-output pairs. The method presented is an evolution of the "peeling algorithm" of L. Lin, J. Lu, and L. Ying, "Fast construction of hierarchical matrix representation from matrix-vector multiplication," JCP, 230(10), 2011. For the case of uniform trees, the new method substantially reduces the pre-factor of the original peeling algorithm. More significantly, the new technique leads to dramatic acceleration for many non-uniform trees since it constructs sample vectors that are optimized for a given tree. The algorithm is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space, such as a boundary integral equation defined on a surface in three dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_03406 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Randomized Compression of Rank-Structured Matrices Accelerated with Graph Coloring Levitt, James Martinsson, Per-Gunnar Numerical Analysis 65N22, 65N38, 15A23, 15A52 A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the assumption that algorithms for rapidly applying $A$ and $A^{*}$ to vectors are available. The algorithm analyzes the hierarchical tree that defines the rank structure using graph coloring algorithms to generate a set of random test vectors. The matrix is then applied to the test vectors, and in a final step the matrix itself is reconstructed by the observed input-output pairs. The method presented is an evolution of the "peeling algorithm" of L. Lin, J. Lu, and L. Ying, "Fast construction of hierarchical matrix representation from matrix-vector multiplication," JCP, 230(10), 2011. For the case of uniform trees, the new method substantially reduces the pre-factor of the original peeling algorithm. More significantly, the new technique leads to dramatic acceleration for many non-uniform trees since it constructs sample vectors that are optimized for a given tree. The algorithm is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space, such as a boundary integral equation defined on a surface in three dimensions. |
| title | Randomized Compression of Rank-Structured Matrices Accelerated with Graph Coloring |
| topic | Numerical Analysis 65N22, 65N38, 15A23, 15A52 |
| url | https://arxiv.org/abs/2205.03406 |