A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such a 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 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 $4\times 10^6$ by $2\times 10^6$). These experiments demonstrate that this method is especially useful for large ill-conditioned problems with multiple right-hand sides.

翻译：本文提出一种直接求解器，用于求解涉及非均匀离散傅里叶变换矩阵的超定线性系统。此类矩阵可转化为具有分层低秩结构的柯西型形式。我们阐释了该矩阵的秩结构，并证明相关子矩阵的秩仅随矩阵列数呈对数增长。基于此分析，发展了一种快速秩结构分层近似方法，并针对此类系统及相关系统构建了分层最小二乘求解器。所得结果为非均匀离散变换的求逆提供了一种直接方法，其计算复杂度与问题的自由度呈近线性关系。在一维第二类（即正变换）求逆的基准测试中，针对不同的条件数与问题规模（最高达$4\times 10^6$行、$2\times 10^6$列），我们将该求解器与多种迭代求解器及直接求解器进行了对比。实验表明，该方法对多右端项的大规模病态问题尤为有效。