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The singular value decomposition (SVD) of a matrix is a powerful tool formany matrix computation problems. In this paper, we consider generalizing thestandard SVD to analyze and compute the regularized solution of linearill-posed problems that arise from discretizing the first kind Fredholmintegral 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 theweighted SVD (WSVD) of a matrix under the $M$-inner product. Several importantapplications of WSVD, such as low-rank approximation and solving the leastsquares problems with minimum $\|\cdot\|_M$-norm, are studied. We propose theweighted Golub-Kahan bidiagonalization (WGKB) to compute several dominant WSVDcomponents and a corresponding weighted LSQR algorithm to iteratively solve theleast squares problem. All the above tools and methods are used to analyze andsolve linear ill-posed problems with the regularizer $\|x\|_{M}^2$. AWGKB-based subspace projection regularization method is proposed to efficientlycompute a good regularized solution, which can incorporate the priorinformation about $x$ encoded by the regularizer $\|x\|_{M}^2$. Severalnumerical experiments are performed to illustrate the fruitfulness of ourmethods.

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