Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

Ioannis K Argyros,Santhosh George,Yeol Je Cho

doi:10.1186/1687-2770-2013-114

Abstract

AbstractIn this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).MSC:65F22, 65J15, 65J22, 65M30, 47A52.

Highlights

1 Introduction In this paper, we are interested in obtaining a stable approximate solution for a nonlinear ill-posed operator equation of the form
In the Lavrentiev regularization, the approximate solution is obtained as a solution of the equation
In Section, we consider some basic assumptions required throughout the paper

Summary

Introduction

They showed that xδkδ → xas δ → under the following assumptions: ( ) There exists L > such that F (x) – F (y) ≤ L x – y for all x, y ∈ D(F); ( ) There exists p > such that αk – αk+ ≤ p αk αk+ In Section , we consider some basic assumptions required throughout the paper. There are many classes of operators satisfying Assumption .

Results

Conclusion