A SVD accelerated kernel-independent fast multipole method and its application to BEM

TL;DR

This paper introduces an SVD-based acceleration for the kernel-independent fast multipole method, significantly improving efficiency and memory usage, and applies it to boundary element methods with demonstrated performance gains.

Contribution

The paper develops an SVD-based acceleration technique for all translations in KIFMM, enhancing efficiency and reducing memory, and applies it to accelerate boundary element methods.

Findings

Reduces about 40% of iteration time.

Decreases memory requirements by 25%.

Demonstrates improved performance in three numerical examples.

Abstract

The kernel-independent fast multipole method (KIFMM) proposed in [1] is of almost linear complexity. In the original KIFMM the time-consuming M2L translations are accelerated by FFT. However, when more equivalent points are used to achieve higher accuracy, the efficiency of the FFT approach tends to be lower because more auxiliary volume grid points have to be added. In this paper, all the translations of the KIFMM are accelerated by using the singular value decomposition (SVD) based on the low-rank property of the translating matrices. The acceleration of M2L is realized by first transforming the associated translating matrices into more compact form, and then using low-rank approximations. By using the transform matrices for M2L, the orders of the translating matrices in upward and downward passes are also reduced. The improved KIFMM is then applied to accelerate BEM. The performance of the proposed algorithms are demonstrated by three examples. Numerical results show that, compared with the original KIFMM, the present method can reduce about 40% of the iterating time and 25% of the memory requirement.