Generalizing the SVD of a matrix under nonstandard 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 a generalization of 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‖M2:=x⊤Mx should be exploited, where M is symmetric positive definite. To handle this regularizer, we use the weighted SVD (WSVD) of a matrix under the M-inner product. Several important applications of the WSVD, such as low-rank approximation and solving the least squares problems with minimum ‖⋅‖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‖M2. Several WGKB based iterative regularization and hybrid regularization methods are proposed to compute a good regularized solution, which can incorporate the prior information about x encoded by the regularizer ‖x‖M2. Several numerical experiments are performed to illustrate the fruitfulness of our methods.