AI 中文总结
针对大规模酉可对角化线性反问题求解的计算难题，提出一种谱域伪逆构造方法，通过解析SVD与谱域滤波构建稳定伪逆，等价于零阶Tikhonov正则化且收敛于Moore--Penrose广义逆，为这类问题提供高效求解途径。
AI 中文摘要
线性反问题广泛存在于地球物理学、信号处理、图像复原和医学成像领域。在数学上，它们可以表述为$Gm = d$。当$G$为病态矩阵或奇异矩阵时，该问题就成为不适定问题，需要采用正则化方法或广义逆方法来获得稳定解。然而，对于大规模线性反问题，这两类方法都面临着显著的计算困难。\n本文针对一类可通过酉矩阵对角化的线性反问题，建立了一种谱域伪逆构造方法。其核心思想是从矩阵的酉对角化结构出发，直接在变换域中构造稳定的伪逆算子。我们首先给出了这类矩阵的解析奇异值分解（Singular Value Decomposition, SVD），阐明了谱分解与SVD之间的对应关系。在此基础上，我们定义了谱域正则化滤波因子，构造了稳定的谱域伪逆算子，并证明了其有界稳定性以及收敛于Moore--Penrose广义逆的一致性。\n该构造在数值上等价于零阶Tikhonov正则化，且当$\alpha\to 0^+$时收敛于Moore--Penrose广义逆，但其方法论路径与两者均不相同。本文揭示了一类结构化矩阵的稳定广义逆可以直接从其谱分解构造得到，为大规模结构化反问题提供了一种高效的求解方法。当酉矩阵取为离散傅里叶变换矩阵时，傅里叶变换情形是该方法的一个特例。
英文摘要
Linear inverse problems are prevalent in geophysics, signal processing, image restoration, and medical imaging. Mathematically, they can be formulated as $Gm = d$. When $G$ is ill-conditioned or singular, the problem becomes ill-posed and requires regularization methods or generalized inverse methods for a stable solution. However, both types of methods encounter significant computational difficulties for large-scale linear inverse problems. In this paper, we establish a spectral-domain pseudo-inverse construction method for a class of linear inverse problems that can be diagonalized by unitary matrices. The core idea is to construct a stable pseudo-inverse operator directly in the transform domain, starting from the unitary diagonalization structure of the matrix. We first provide the analytic Singular Value Decomposition (SVD) of this class of matrices, clarifying the correspondence between spectral decomposition and SVD. On this basis, we define spectral-domain regularization filtering factors, construct a stable spectral-domain pseudo-inverse operator, and prove its bounded stability as well as its consistency in converging to the Moore--Penrose generalized inverse. This construction is numerically equivalent to zeroth-order Tikhonov regularization and converges to the Moore--Penrose generalized inverse as $α\to 0^+$, but its methodological path differs from both. This paper reveals that the stable generalized inverse of a class of structured matrices can be directly constructed from their spectral decomposition, providing an efficient solution method for large-scale structured inverse problems. The Fourier transform case is a special instance of this method when the unitary matrix is taken as the discrete Fourier transform matrix.
