Regularization with randomized SVD for large-scale discrete inverse problems

In this paper, we propose an algorithm for solving the large-scale discrete ill-conditioned linear problems arising from the discretization of linear or nonlinear inverse problems. The algorithm combines some existing regularization techniques and regularization parameter choice rules with a randomized singular value decomposition (SVD), so that only much smaller scale systems are needed to solve, instead of the original large-scale regularized system. The algorithm can directly apply to some existing regularization methods, such as the Tikhonov and truncated SVD methods, with some popular regularization parameter choice rules such as the L-curve, GCV function, quasi-optimality and discrepancy principle. The error of the approximate regularized solution is analyzed and the efficiency of the method is well demonstrated by the numerical examples.