Tikhonov replacement functionals for iteratively solving nonlinear operator equations

Abstract

We shall be concerned with the construction of Tikhonov-based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On the basis of the minimizers of the replacement functional, we introduce an iterative algorithm that converges towards a critical point of the Tikhonov functional, and under additional assumptions for the nonlinear operator F, to a global minimizer. Combining our iterative strategy with an appropriate parameter selection rule, we obtain convergence and convergence rates. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (single photon emission computerized tomography) problem.