Extrapolation of tikhonov regularization method*

Abstract

We consider regularization of linear ill-posed problem Au = f with noisy data fδ, {pipe}{pipe}fδ - f{pipe}{pipe} ≤ δ. The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A)n), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ2/3) versus accuracy O(δ2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over-and underestimation of the noise level. Numerical examples are given. © Vilnius Gediminas Technical University, 2010.