Randomized Algorithms for Large-scale Inverse Problems with General Tikhonov Regularizations

We shall investigate randomized algorithms for solving large-scale linear inverse problems with general Tikhonov regularizations. Our first approach transforms inverse problems of general form into the ones of standard form, then apply randomized algorithms to reduce large-scale systems of standard form to much smaller-scale systems and seek their regularized solutions in combination with some popular choice rules for regularization parameters. Our second approach involves a new random generalized SVD algorithm that can essentially reduce the sizes of the original large-scale ill-posed systems. The reduced systems can provide approximate regularized solutions with about the same accuracy as the ones by the classical generalized SVD, but they are much more stable and much less expensive as they need only to work on problems of much smaller sizes. Numerical results are presented to demonstrate the efficiency and accuracy of the algorithms.