Preconditioning CGNE iteration for inverse problems

The conjugate gradient method applied to the normal equations (CGNE) is known as efficient method for the solution of non-symmetric linear equations. By stopping the iteration according to a discrepancy principle, CGNE can be turned into a regularization method, and thus can be applied to the solution of inverse, in particular, ill-posed problems. We show that CGNE for inverse problems can be further accelerated by preconditioning in Hilbert scales, derive (optimal) convergence rates with respect to data noise, and give tight bounds on the iteration numbers. The theoretical results are illustrated by numerical tests. Copyright (C) 2007 John Wiley & Sons, Ltd.