Extrapolation of Tikhonov Regularization Method

Abstract

Abstract. We consider regularization of linear ill-posed problem Au = f with noisy data f δ , f δ − f ≤ 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(δ 2/3 ) versus accuracy O(δ 2n/(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.

A family of rules for the choice of the regularization parameter in the Lavrentiev method in the case of rough estimate of the noise level of the data

“…Note also that in the case of the smooth solution u a proper linear combination of m Lavrentiev approximations (v˛, if m D 2) with different values of the parameter˛is a more accurate approximation to u than the single Lavrentiev approximation (see [6,7,9,26]). …”

“…The discrepancy of extrapolated approximation (DE rule) ( [6,7,9,26]). Here d.˛/ D kAv˛ Q f k; v˛WD u˛ ru r1 r with r 6 D 1, r > 0 is the extrapolated approximation, b 0 D 1.…”

“…Numerical examples in Section 7 show that the rules from the proposed family work well also in the case the noise level is under-or overestimated many times. Note that extensive numerical comparison of other parameter choice rules can be found in [3,[8][9][10]24]. …”

Heuristic Parameter Choice in Tikhonov Method from Minimizers of the Quasi-Optimality Function

“…max ] the parameter α j * belongs, we take maximum of T over all intervals, 1 ≤ k ≤ K. Using also (11) we obtain the estimate (8).…”

Effective Alternating Direction Optimization Methods for Sparsity-Constrained Blind Image Deblurring

“…During the past several decades, numerous numerical methods have been developed to handle non-blind deconvolution. One of the most popular methods is the Tikhonov regularization [ 16 , 17 ], followed by its various extensions [ 18 , 19 ]. These methods can be easily implemented, but commonly generate over-smoothing effects on the restored images.…”