Superfast direct inversion of the nonuniform discrete Fourier transform via hierarchically semi-separable least squares

TL;DR

This paper introduces a fast, direct hierarchical least-squares solver for inverting nonuniform discrete Fourier transform matrices, achieving nearly linear complexity and outperforming traditional methods on large-scale problems.

Contribution

The paper presents a novel hierarchical approximation method and direct solver for nonuniform DFT matrices, exploiting their low-rank structure for efficient inversion.

Findings

Solver achieves nearly linear complexity.

Outperforms iterative and direct solvers on large problems.

Effective for problems with multiple right-hand sides.

Abstract

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem.This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to 4 x 10^6 by 2 x 10^6). These experiments demonstrate that this method is especially useful for large problems with multiple right-hand sides.