New rule for choice of the regularization parameter in (iterated) tikhonov method

Abstract

We propose a new a posteriori rule for choosing the regularization parameter α in (iterated) Tikhonov method for solving linear ill‐posed problems in Hilbert spaces. We assume that data are noisy but noise level δ is given. We prove that (iterated) Tikhonov approximation with proposed choice of α converges to the solution as δ → 0 and has order optimal error estimates. Under certain mild assumption the quasioptimality of proposed rule is also proved. Numerical examples show the advantage of the new rule over the monotone error rule, especially in case of rough δ.

Key words:

• ill‐posed problem
• (iterated) Tikhonov method
• regularization parameter
• a posteriori rule
• monotone error rule
• convergence
• order optimality

Notes

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