Residual norm behavior for Hybrid LSQR regularization

Abstract

Hybrid LSQR represents a powerful method for regularization of large-scale discrete inverse problems, where ill-conditioning of the model matrix and ill-posedness of the problem make the solutions seriously sensitive to the unknown noise in the data. Hybrid LSQR combines the iterative Golub-Kahan bidiagonalization with the Tikhonov regularization of the projected problem. While the behavior of the residual norm for the pure LSQR is well understood and can be used to construct a stopping criterion, this is not the case for the hybrid method. Here we analyze the behavior of norms of approximate solutions and the corresponding residuals in Hybrid LSQR with respect to the Tikhonov regularization parameter. This helps to understand convergence properties of the hybrid approach. Numerical experiments demonstrate the results in finite precision arithmetic.