A superfast direct inversion method for the nonuniform discrete Fourier transform
Abstract
A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such a matrices can be transformed into a Cauchylike 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 rankstructured hierarchical approximation method based on this analysis is developed, along with a hierarchical leastsquares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is 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 onedimensional typeII (or forward) transform,for a range of condition numbers and problem sizes (up to $4\times 10^6$ by $2\times 10^6$). These experiments demonstrate that this method is especially useful for large illconditioned problems with multiple righthand sides.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.13223
 arXiv:
 arXiv:2404.13223
 Bibcode:
 2024arXiv240413223W
 Keywords:

 Mathematics  Numerical Analysis;
 65T50;
 65F55;
 65F20
 EPrint:
 28 pages, 8 figures