Fast construction of hierarchical matrix representation from matrix-vector multiplication

Authors:
Lin Lin;Jianfeng Lu;Lexing Ying
Affiliations:
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States;Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, United States;Department of Mathematics and ICES, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, United States
Venue:
Journal of Computational Physics
Year:
2011

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Abstract

We develop a hierarchical matrix construction algorithm using matrix-vector multiplications, based on the randomized singular value decomposition of low-rank matrices. The algorithm uses O(logn) applications of the matrix on structured random test vectors and O(nlogn) extra computational cost, where n is the dimension of the unknown matrix. Numerical examples on constructing Green's functions for elliptic operators in two dimensions show efficiency and accuracy of the proposed algorithm.