Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems
In a recent paper [Numer. Funct. Anal. Optimization 27, No. 5--6, 637--665 (2006; Zbl 1101.65054)], the author suggested a multilevel method for solving ill-posed operator equations governed by a compact, weakly sequentially closed mapping \(F\) between two Hilbert spaces. This stable and globally convergent method combines regularization by discretization with an iterative approach. In this paper, optimal convergence rates of Hölder- or logarithmic type are proven under suitable assumptions on a linearization of \(F\) and on the initial guess. Furthermore, numerical aspects of the minimization problems appearing on each level of discretization are studied, and it is shown that it suffices to consider Karush-Kuhn-Tucker points rather than global minimizers. Finally, numerical tests are carried out for a nonlinear integral equation with an analytic kernel.
scientific article; zbMATH DE number 5306881
scientific article