The singular value decomposition (SVD) of a matrix is a powerful tool for many matrix computation problems. In this paper, we consider generalizing the standard SVD to analyze and compute the regularized solution of linear ill-posed problems that arise from discretizing the first kind Fredholm integral equations. For the commonly used quadrature method for discretization, a regularizer of the form $\|x\|_{M}^2:=x^TMx$ should be exploited, where $M$ is symmetric positive definite. To handle this regularizer, we give the weighted SVD (WSVD) of a matrix under the $M$-inner product. Several important applications of WSVD, such as low-rank approximation and solving the least squares problems with minimum $\|\cdot\|_M$-norm, are studied. We propose the weighted Golub-Kahan bidiagonalization (WGKB) to compute several dominant WSVD components and a corresponding weighted LSQR algorithm to iteratively solve the least squares problem. All the above tools and methods are used to analyze and solve linear ill-posed problems with the regularizer $\|x\|_{M}^2$. A WGKB-based subspace projection regularization method is proposed to efficiently compute a good regularized solution, which can incorporate the prior information about $x$ encoded by the regularizer $\|x\|_{M}^2$. Several numerical experiments are performed to illustrate the fruitfulness of our methods.

翻译：矩阵的奇异值分解（SVD）是解决许多矩阵计算问题的有力工具。本文考虑推广标准SVD，以分析和计算由第一类Fredholm积分方程离散化产生的线性不适定问题的正则化解。针对常用的数值求积离散方法，应采用形如$\|x\|_{M}^2:=x^TMx$的正则化项，其中$M$为对称正定矩阵。为处理该正则化项，本文提出在$M$-内积下的加权SVD（WSVD）。研究了WSVD的若干重要应用，包括低秩逼近和求解具有最小$\|\cdot\|_M$-范数的最小二乘问题。我们提出加权Golub-Kahan双对角化（WGKB）方法以计算若干主导WSVD成分，并设计相应的加权LSQR算法迭代求解最小二乘问题。上述所有工具与方法均用于分析并求解带正则化项$\|x\|_{M}^2$的线性不适定问题。进一步提出基于WGKB的子空间投影正则化方法，可高效计算包含由$\|x\|_{M}^2$编码的先验信息$x$的良好正则化解。数值实验验证了所提出方法的有效性。