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A note on logarithmic convergence rates for nonlinear Tikhonov regularization. (English) Zbl 1154.65042

A nonlinear ill-posed operator equation F(x)=y, from the domain D(F)X of the Hilbert space X into a Hilbert space Y, is considered in case of only noisy data y δ are available, with the assumption y-yδ, δ>0. The Tikhonov regularization method consists in using the Tikhonov functional

J α (x)=F(x)-y δ 2 +αx-x 0 2 ·

The convergence rates for this type of regularization under a mild regularity assumption on the solution, namely source conditions of logarithmic type, are proved. For the choice of the regularization parameter a priori or a posteriori strategies according to the discrepancy can be used. Restrictions on the nonlinearity of the forward operator are made unless the initial error is sufficiently smooth.

65J15Equations with nonlinear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
47J06Nonlinear ill-posed problems