A novel modified TRSVD method for large-scale linear discrete ill-posed problems

Abstract

The truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. This method replaces the matrix A by the closest matrix Ak of low rank k, and then computes the minimal norm solution of the linear system of equations with a rank-deficient matrix so obtained. The modified TSVD (MTSVD) method improves the TSVD method, by replacing A by a matrix that is closer to A than Ak in a unitarily invariant matrix norm and has the same spectral condition number as Ak. Approximations of the SVD of a large matrix A can be computed quite efficiently by using a randomized SVD (RSVD) method. This paper presents a novel modified truncated randomized singular value decomposition (MTRSVD) method for computing approximate solutions to large-scale linear discrete ill-posed problems. The rank, k, is determined with the aid of the discrepancy principle, but other techniques for selecting a suitable rank also can be used. Numerical examples illustrate the effectiveness of the proposed method and compare it to the truncated RSVD method.