A Multiscale Butterfly Algorithm for Multidimensional Fourier Integral Operators

TL;DR

This paper introduces a simple, efficient multiscale butterfly algorithm for computing multidimensional Fourier integral operators with quasi-linear complexity, avoiding coordinate transformations and demonstrating practical advantages in 2D and 3D.

Contribution

The paper develops a novel multiscale butterfly algorithm for FIOs that reduces computational cost and complexity compared to previous methods, without requiring coordinate transformations.

Findings

Achieves quasi-linear operation complexity

Demonstrates linear memory complexity

Shows practical efficiency in 2D and 3D numerical examples

Abstract

This paper presents an efficient multiscale butterfly algorithm for computing Fourier integral operators (FIOs) of the form $(\mathcal{L} f)(x) = \int_{R^d}a(x,\xi) e^{2\pi \i \Phi(x,\xi)}\hat{f}(\xi) d\xi$, where $\Phi(x,\xi)$ is a phase function, $a(x,\xi)$ is an amplitude function, and $f(x)$ is a given input. The frequency domain is hierarchically decomposed into a union of Cartesian coronas. The integral kernel $a(x,\xi) e^{2\pi \i \Phi(x,\xi)}$ in each corona satisfies a special low-rank property that enables the application of a butterfly algorithm on the Cartesian phase-space grid. This leads to an algorithm with quasi-linear operation complexity and linear memory complexity. Different from previous butterfly methods for the FIOs, this new approach is simple and reduces the computational cost by avoiding extra coordinate transformations. Numerical examples in two and three dimensions are provided to demonstrate the practical advantages of the new algorithm.