Tikhonov regularization and a posteriori rules for solving nonlinear ill posed problems

Besides a priori parameter choice we study a posteriori rules for choosing the regularization parameter α in the Tikhonov regularization method for solving nonlinear ill posed problems F (x) = y, namely a rule 1 of Scherzer et al (Scherzer O, Engl H W and Kunisch K 1993 SIAM J. Numer. Anal. 30 1796–838) and a new rule 2 which is a generalization of the monotone error rule of Tautenhahn and Hamarik (Tautenhahn U and Hamarik U 1999 Inverse Problems 15 1487–505) to the nonlinear case. We suppose that instead of y there are given noisy data yδ satisfying |y − yδ| ≤ δ with known noise level δ and prove that rule 1 and rule 2 yield order optimal convergence rates O(δp/(p+1)) for the ranges p (0, 2] and p (0, 1], respectively. Compared with foregoing papers our order optimal convergence rate results have been obtained under much weaker assumptions which is important in engineering practice. Numerical experiments verify some of the theoretical results.

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