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Journal of Computational and Applied Mathematics
Article . 2024 . Peer-reviewed
License: Elsevier TDM
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Article . 2024
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https://dx.doi.org/10.48550/ar...
Article . 2022
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Randomized compression of rank-structured matrices accelerated with graph coloring

Authors: James Levitt; Per-Gunnar Martinsson;

Randomized compression of rank-structured matrices accelerated with graph coloring

Abstract

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.

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Keywords

Random matrices (algebraic aspects), HODLR matrix, Numerical linear algebra, Randomized algorithms, hierarchically block separable matrix, Boundary element methods for boundary value problems involving PDEs, Numerical Analysis (math.NA), Numerical solution of discretized equations for boundary value problems involving PDEs, Direct numerical methods for linear systems and matrix inversion, Factorization of matrices, Computational methods for sparse matrices, hierarchically semiseparable matrix, rank-structured matrices, randomized approximation of matrices, FOS: Mathematics, Mathematics - Numerical Analysis, 65N22, 65N38, 15A23, 15A52, fast direct solver

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Top 10%
Average
Average
Green