Multidimensional Butterfly Factorization

TL;DR

This paper presents the multidimensional butterfly factorization, a data-sparse method for approximating certain kernel matrices efficiently, with algorithms tailored for different kernel evaluation scenarios and extensions for singularities.

Contribution

It introduces the multidimensional butterfly factorization, extending it to handle singularities in Fourier integral operators, and provides efficient construction algorithms.

Findings

Achieves $ ext{O}( ext{log} N)$ sparse matrix product approximation.

Extends the factorization to Fourier integral operators with singularities.

Demonstrates efficiency through numerical experiments.

Abstract

This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size $N\times N$ with a product of $\O(\log N)$ sparse matrices, each of which contains $\O(N)$ nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in $\O(1)$ operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.