Use of extrapolation in regularization methods

Abstract

Extrapolation is a well-known tool for increasing the accuracy of approximation methods. We consider extrapolation of Tikhonov and Lavrentiev methods and iterated variants of these methods for solving linear ill-posed problems in Hilbert space. For extrapolated approximation we take the linear combination of n ≥ 2 approximations of the original method with different parameters and with proper coefficients, guaranteeing for extrapolated method higher qualification than in original method. Extrapolated approximation can be used for approximation to solution of the equation or for construction of a posteriori rules for choice of the regularization parameter in original method. As shown, extrapolating n Tikhonov or Lavrentiev approximations gives the same approximation as one gets in nonstationary implicit iterative method after n iterations with the same parameters. If the solution is smooth and δ is noise level of data, a proper choice of n = n (δ) guarantees for extrapolated Tikhonov approximation accuracy versus accuracy of Tikhonov approximation. Note that in a posteriori parameter choice in Tikhonov method often several approximations with different parameters are computed and then computation of their linear combination is an easy task.