On the Solution of Linear Ill-posed Problems on the Basis of a Modified Quasioptimality Criterion

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)) and instead of f noisy data fδ with ‖fδ − f‖ ≤ δ are available, maximal guaranteed accuracy of Tikhonov and Lavrentiev approximations is O(δ) and O(δ), respectively, versus accuracy O(δ) and O(δ) 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∗ ∈ 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.