Author(s)

MINGJUAN LI, WENZHI XU, ZHUOJIA FU

Abstract

This paper presents a novel approach to solving large-scale linear algebra problems by integrating hierarchical matrices (H-matrices) with a semi-analytic meshless collocation method (singular boundary method). The singular boundary method eliminates the need for mesh generation by using a set of discrete nodes to approximate the solution of partial differential equations. However, this approach often leads to dense linear systems that are computationally expensive and memory-intensive. To address this challenge, we employ H-matrices, which decompose these dense matrices into hierarchical sub-blocks that can often be approximated by low-rank matrices. This decomposition significantly reduces computational complexity from the conventional to or lower, depending on the matrix structure. The implementation of these data-sparse representations significantly enhances the efficiency of numerical computations by reducing both complexity and storage requirements traditionally associated with matrix operations. Our results demonstrate that the combination of H-matrices with the singular boundary method not only reduces memory requirements but also accelerates matrix operations, such as matrix-vector multiplication and inversion. The validity of the proposed method for solving the Laplace equation and the Helmholtz equation is verified by benchmark examples, highlighting its potential for large-scale engineering problems.

Keywords

meshless collocation method, singular boundary method, hierarchical matrices, large-scale linear algebra, low-rank approximation