Toward Global Convergence for Strongly Nonlinear Ill-Posed Problems via a Regularizing Multilevel Approach

Abstract

In this paper, we propose a multilevel method for solving nonlinear ill-posed operator equations

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, which is 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. Moreover, this approach enables one to relax restrictions on the nonlinearity of the forward operator, as were used in previous work on regularization methods for nonlinear ill-posed problems.

Keywords:

• Multilevel method
• Nonlinear ill-posed problems
• Regularization

AMS Subject Classification:

• 65J15
• 65J20
• 47J06
• 65R30

ACKNOWLEDGMENTS

This work was supported by the German Science Foundation DFG under grant Ka 1778/1 within the junior research group Inverse Problems in Piezoelectricity.