Extrapolation of Tikhonov and Lavrentiev regularization methods

We consider solution of linear ill-posed problem Au = f by Tikhonov method and by Lavrentiev method. For increasing the qualification and accuracy of these methods we use extrapolation, taking for the approximate solution linear combination of n ≥ 2 approximations of Tikhonov or Lavrentiev methods with different parameters and with proper coefficients. If the solution u* belongs to R((A*A)n) and instead of f noisy data fδ with ||fδ - f|| ≥ δ are available, maximal guaranteed accuracy of Tikhonov and Lavrentiev approximations is O(δ2/3) and O(δ1/2), respectively, versus accuracy O(δ2n/(2n+1)) and O(δn/(n+1)) of corresponding extrapolated approximations. We propose several new rules for a posteriori choice of the regularization parameter, including modifications of the monotone error rule. Extensive numerical experiments show that in case u* ε R(A*) the extrapolated Tikhonov approximation with a posteriori parameter choice (not using any smoothness information) is typically more accurate than Tikhonov approximation with optimal parameter.