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

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.