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

Haibo Li

Applied Mathematics and Computation, 2026, vol. 508, issue C

Abstract: 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.

Keywords: Fredholm integral equation; Ill-posed problems; Weighted singular value decomposition; Subspace projection regularization; Weighted Golub-Kahan bidiagonalization; Weighted LSQR (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:508:y:2026:i:c:s0096300325003340

DOI: 10.1016/j.amc.2025.129608

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