Conjugate gradient for ill-posed problems: regularization by preconditioning, preconditioning by regularization

Abstract

This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive semi-definite operator, a powerful Ritz analysis allows one to estimate at negligible computational cost the solution for any Tikhonov's weight. This enhanced linear solver is applied to the boundary data completion problem and as the inner solver for the optical flow estimator.