Extrapolation of Tikhonov regularization method

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.

Keywords:

• ill‐posed problems
• regularization
• Tikhonov method
• extrapolation
• noise level
• regularization parameter choice
• balancing principle
• monotone error rule
• rule R2

Notes

This work was supported by the Estonian Science Foundation, Research Grant No. 7489.