A Hybrid LSMR Algorithm for Large-Scale Tikhonov Regularization

Abstract

We develop a hybrid iterative approach for computing solutions to large-scale ill-posed inverse problems via Tikhonov regularization. We consider a hybrid LSMR algorithm, where Tikhonov regularization is applied to the LSMR subproblem rather than the original problem. We show that, contrary to standard hybrid methods, hybrid LSMR iterates are not equivalent to LSMR iterates on the directly regularized Tikhonov problem. Instead, hybrid LSMR leads to a different Krylov subspace problem. We show that hybrid LSMR shares many of the benefits of standard hybrid methods such as avoiding semiconvergence behavior. In addition, since the regularization parameter can be estimated during the iterative process, it does not need to be estimated a priori, making this approach attractive for large-scale problems. We consider various methods for selecting regularization parameters and discuss stopping criteria for hybrid LSMR, and we present results from image processing.