Tikhonov regularization and a posteriori rules for solving nonlinear ill posedproblems

Abstract

Besides a priori parameter choice we study a posteriori rules for choosing the regularizationparameter αin the Tikhonov regularization method for solving nonlinear ill posed problemsF (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 themonotone error rule of Tautenhahn and Hämarik (Tautenhahn U and Hämarik U 1999Inverse Problems 15 1487–505) to the nonlinear case. We suppose that instead ofy there are givennoisy data yδsatisfying |y − yδ| ≤ δ with known noiselevel δand prove that rule 1 and rule 2 yield order optimal convergence rates O(δp/(p+1)) for theranges p ∊ (0, 2]and p ∊ (0, 1],respectively. Compared with foregoing papers our order optimal convergence rateresults have been obtained under much weaker assumptions which is important inengineering practice. Numerical experiments verify some of the theoretical results.