Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems

Abstract

In this paper, we prove convergence rates for a previously [22] proposed multilevel method for solving nonlinear ill-posed operator equations F(x) = y. By minimizing the distance to some initial guess under the constraint of a discretized version of the operator equation for different levels of discretization, we define a sequence of regularized approximations to the exact solution, that in [22] had been shown to be stable and convergent for arbitrary initial guess, and can be computed via a multilevel procedure that altogether yields a globally convergent method. In the present paper we prove optimal logarithmic and Ḧolder type convergence rates under respective source conditions. Moreover we provide a tool for possible numerical solution strategies for the minimization problem on each level of discretization by providing an exact penalty function derived via an augmented Lagrangian approach. © 2008 Rocky Mountain Mathematics Consortium.