Iterative Regularization and Generalized Discrepancy Principle for Monotone Operator Equations

Abstract

In this paper, we propose two novel algorithms for solving a nonlinear ill-posed operator equation F(x) = f _δ, ‖f − f _δ‖ ≤ δ, in a real Hilbert space H under the following basic assumption:

The first algorithm is a combination of iteratively regularized Newton's scheme (IRNS)

with a posteriori stopping rule

The second algorithm is a modified version of IRNS with simultaneous updates of the operator [F′(x _n ) + ϵ _n I]⁻¹:

combined with (1). A rigorous theoretical analysis of both methods is conducted.

Keywords:

• Discrepancy principle
• Ill-posed problem
• Regularization

AMS Subject Classification:

• 47A52
• 65F22

ACKNOWLEDGMENTS

The authors would like to express their special gratitude to Dr. Zuhair Nashed for his helpful comments and corrections to the preliminary version of this paper.

This work was supported by RFFI grant (06-01-00282) and NSF grant (DMS-0207050).