Fast construction of hierarchical matrix representation from matrix-vector multiplication

TL;DR

This paper presents a fast, randomized algorithm for constructing hierarchical matrix representations from matrix-vector multiplications, significantly reducing computational cost while maintaining accuracy, demonstrated through elliptic operator Green's functions.

Contribution

The authors introduce a novel hierarchical matrix construction method leveraging randomized SVD and structured random vectors, achieving logarithmic application complexity.

Findings

Efficient construction of hierarchical matrices from matrix-vector products.

Numerical validation on elliptic operators shows high accuracy.

Algorithm scales well with matrix size, with reduced computational cost.

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 $\mathcal{O}(\log n)$ applications of the matrix on structured random test vectors and $\mathcal{O}(n \log n)$ 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.