Tikhonov regularization with MTRSVD method for solving large-scale discrete ill-posed problems

Abstract

Regularization is possibly the most popular method for solving discrete ill-posed problems, whose solution is less sensitive to the error in the observed vector in the right hand than the original solution. This paper presents a new modified truncated randomized singular value decomposition (TR-MTRSVD) method for large Tikhonov regularization in standard form. The proposed TR-MTRSVD algorithm introduces the idea of randomized algorithm into the improved truncated singular value decomposition (MTSVD) method to solve large Tikhonov regularization problems. The approximation matrix Ãℓ produced by randomized SVD is replaced by the closest matrix Ãk̃ in a unitarily invariant matrix norm with the same spectral condition number. The regularization parameters are determined by the discrepancy principle. Numerical examples show the effectiveness and efficiency of the proposed TR-MTRSVD algorithm for large Tikhonov regularization problems.