Hybrid LSMR algorithms for large-scale general-form regularization

Abstract

The hybrid LSMR algorithm is proposed for large-scale general-form regularization. It is based on a Krylov subspace projection method where the matrix $A$ is first projected onto a subspace, typically a Krylov subspace, which is implemented via the Golub-Kahan bidiagonalization process applied to $A$, with starting vector $b$. Then a regularization term is employed to the projections. Finally, an iterative algorithm is exploited to solve a least squares problem with constraints. The resulting algorithms are called the {hybrid LSMR algorithm}. At every step, we exploit LSQR algorithm to solve the inner least squares problem, which is proven to become better conditioned as the number of $k$ increases, so that the LSQR algorithm converges faster. We prove how to select the stopping tolerances for LSQR in order to guarantee that the regularized solution obtained by iteratively computing the inner least squares problems and the one obtained by exactly computing the inner least squares problems have the same accuracy. Numerical experiments illustrate that the best regularized solution by the hybrid LSMR algorithm is as accurate as that by JBDQR which is a joint bidiagonalization based algorithm.