On the Compression of Translation Operator Tensors in FMM-FFT-Accelerated SIE Simulators via Tensor Decompositions

TL;DR

This paper introduces tensor decomposition techniques to compress translation operator tensors in FMM-FFT-accelerated SIE simulations, significantly reducing memory usage with minimal computational overhead.

Contribution

It applies Tucker, hierarchical Tucker, and tensor train decompositions to efficiently compress translation tensors in FMM-FFT SIE simulators, enhancing memory efficiency.

Findings

H-Tucker yields maximum memory savings for 4D tensors.

Tucker decomposition introduces minimal computational overhead.

All methods significantly reduce memory requirements in practical scenarios.

Abstract

Tensor decomposition methodologies are proposed to reduce the memory requirement of translation operator tensors arising in the fast multipole method-fast Fourier transform (FMM-FFT)-accelerated surface integral equation (SIE) simulators. These methodologies leverage Tucker, hierarchical Tucker (H-Tucker), and tensor train (TT) decompositions to compress the FFT'ed translation operator tensors stored in three-dimensional (3D) and four-dimensional (4D) array formats. Extensive numerical tests are performed to demonstrate the memory saving achieved by and computational overhead introduced by these methodologies for different simulation parameters. Numerical results show that the H-Tucker-based methodology for 4D array format yields the maximum memory saving while Tucker-based methodology for 3D array format introduces the minimum computational overhead. For many practical scenarios, all methodologies yield a significant reduction in the memory requirement of translation operator tensors while imposing negligible/acceptable computational overhead.