Approximate inversion of discrete Fourier integral operators

TL;DR

This paper presents a novel quasi-linear time method for approximating the inverse of discrete Fourier integral operators using hierarchical and butterfly factorizations, enabling efficient direct solving and preconditioning.

Contribution

It introduces a new approximate inverse factorization technique combining butterfly and hierarchical interpolative factorizations for Fourier integral operators.

Findings

Achieves approximate inverse in quasi-linear time.

Demonstrates effectiveness on 1D and 2D Fourier integral operators.

Provides a new tool for fast direct solvers and preconditioners.

Abstract

This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral operator is obtained as a product of $O(\log N)$ sparse matrices of size $N\times N$. The resulting approximate inverse factorization can be used as a direct solver or as a preconditioner. Numerical examples on 1D and 2D Fourier integral operators, including a generalized Radon transform, demonstrate the performance of this new approach.