Hierarchical Tucker Low-Rank Matrices: Construction and Matrix-Vector Multiplication

TL;DR

This paper introduces the hierarchical Tucker low-rank (HTLR) matrix, a novel approach for efficiently approximating non-oscillatory kernel functions with linear complexity, enabling fast matrix-vector multiplication and broad applicability.

Contribution

The paper presents the construction and application algorithms for HTLR matrices, combining hierarchical matrices with Tucker low-rank blocks for improved efficiency and versatility.

Findings

Achieves linear complexity in matrix construction

Demonstrates efficient matrix-vector multiplication with quasi-linear complexity

Performs well in memory usage and runtime in numerical experiments

Abstract

In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker low-rank blocks. Using high-dimensional interpolation as well as tensor contractions, algorithms for the construction and matrix-vector multiplication of HTLR matrices are proposed admitting linear and quasi-linear complexities respectively. Numerical experiments demonstrate that the HTLR matrix performs well in both memory and runtime. Furthermore, the HTLR matrix can also be applied on quasi-uniform grids in addition to uniform grids, enhancing its versatility.