Generalizing the SVD of a matrix under non-standard inner product and its applications to linear ill-posed problems

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.