Preconditioning CGNE-iteration for inverse problems

The conjugate gradient method applied to the normal equations (CGNE) is known as one of the most efficient methods 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. We show that CGNE can be accelerated by preconditioning in Hilbert scales, derive (optimal) convergence rates with respect to data noise for the preconditioned method, and illustrate the theoretical results by numerical tests.