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.
翻译:本文介绍了可适用于准线性时间的离散的Fourier整体运营商反转的乘数。乘数的起始点是接近操作商的蝴蝶因子化。接着,为由Fourier整体运营商及其副作用组成的隐士矩阵构造了一个等级矩阵表。这个表示点与基于等级间因数化的新的算法相逆。通过将这两个因数合并,Fourier整体运营商的近似反向因数作为美元(N)的零散基体($N)的产物,以N=N=乘数开始。由此产生的近似反因数化可用作直接的解算器或作为先决条件。 1D 和 2D Fourier整体运营商的数值示例,包括一个通用的拉松变,显示了这一新方法的绩效。