A Superfast Direct Solver for Type-III Inverse Nonuniform Discrete Fourier Transform

TL;DR

The paper introduces a superfast direct solver for the Type-III INUDFT by approximating the NUDFT matrix A as A ≈ B H, where B is a Type-II NUDFT matrix and H is compressible to an HSS matrix. It leverages a randomized HSS construction, a URV factorization, and the fast Type-II INUDFT solver to deliver a quasi-linear pipeline (Construction, Factorization, Solution) that can function as a direct solver or a preconditioner. A theoretical error bound under i.i.d. uniform sampling is provided, and extensive numerical experiments demonstrate robust performance and favorable scaling under both perturbation-based and random sampling scenarios. The approach promises significant speedups for inverse NUDFT tasks in scientific computing and related applications.

Abstract

The nonuniform discrete Fourier transform (NUDFT) and its inverse are widely used in various fields of scientific computing. In this article, we propose a novel superfast direct inversion method for type-III NUDFT. The proposed method approximates the type-III NUDFT matrix as a product of a type-II NUDFT matrix and an HSS matrix, where the type-II NUDFT matrix is further decomposed into the product of an HSS matrix and an uniform discrete Fourier transform (DFT) matrix as in [Wilber, Epperly, and Barnett, SIAM Journal on Scientific Computing, 47(3):A1702-A1732, 2025]. This decomposition enables both the forward application and the backward inversion to be accomplished with quasi-linear complexity. The fast inversion can serve as a high-accuracy direct solver or as an efficient preconditioner. Additionally, we provide an error bound for the approximation under specific sample distributions. Numerical results are presented to verify the relevant theoretical properties and demonstrate the efficiency of the proposed methods.

Figures (7)

• Figure 2.1: An illustration of the URV factorization. (A) Introducing zeros into off-diagonal columns. (B) QR factorization on diagonal blocks.
• Figure 4.1: Approximation error of a type-III NUDFT matrix of size $4096 \times 1024$ for various $R$ and $\alpha$. Left: Approximation error with respect to $R$. Right: Approximation error with respect to $\alpha$. The upper bound is given by $\sqrt{2} / (\pi \sqrt{R - 3 / 2})$ (see\ref{['eq:nudft3_fast_approx_error']}).
• Figure 4.2: CPU times on different stages and relative residual of the proposed type-III INUDFT direct solver under the Perturbation-Perturbation case. The perturbation size is chosen to be $10^{-7}$, $10^{-4}$, $0.1$ and $0.4$. The green dotted lines represent reference complexity of corresponding stages.
• Figure 4.3: The proportion of different parts in the construction stage. The yellow, red and blue parts represent the fADI-construction of $\boldsymbol{B}_{\mathrm{fast}}$, the URV factorization of $\boldsymbol{\tilde{B}}_{\mathrm{HSS}}$ and the black-box compression of $\boldsymbol{H}_{\mathrm{HSS}}$ respectively. Left: Perturbation-Perturbation case. Right: Random-Perturbation case. Under both cases the perturbation of frequencies $\alpha = 0.4$ and $\beta = 0.4$. The column size $N = 2^n$.
• Figure 4.4: CPU times on different stages and relative residual of the proposed type-III INUDFT direct solver under Random-Perturbation case. The perturbation size is chosen to be $10^{-7}$, $10^{-4}$, $0.1$ and $0.4$.

Theorems & Definitions (2)

• Definition 2.1: Xi_Xia_Cauley_Balakrishnan_2014
• Proposition 3.1