The Modified Truncated SVD Method for Regularization in General Form

The truncated singular value decomposition (SVD) method is useful for solving the standard-form regularization problem: min || x||₂ subject to min ||A x - b||₂. This paper presents a modification of the truncated SVD method, which solves the more general problem: min ||L x||₂ subject to min ||A x - b||₂, where L is a general matrix with full row rank. The extra work, associated with the introduction of the matrix L, is dominated by a QR-factorization of a matrix with dimensions smaller than those of L. In order to determine the optimal solution, it is often necessary to compute a sequence of regularized solutions, and it is shown how this can be accomplished with little extra computational effort. Finally, the new method is illustrated with an example from helioseismology.