Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions

Abstract

In this paper we improve existing convergence and convergence rate results for the iteratively regularized Gauss-Newton method in two respects: First we show optimal rates of convergence under general source conditions, and second we assume that the linearized equations are solved only approximately in each Newton step. The latter point is important for large scale problems where the linearized equation can often only be solved iteratively, e.g. by the conjugate gradient method.

Iterative Solution Methods

“…Convergence and optimal rates for the iteratively regularized Gauss-Newton method were established in [82] under a general source condition of the form…”

Complexity analysis of the iteratively regularized Gauss–Newton method with inner CG-iteration

“…The proof of a) follows along the lines of the proof of Theorem 4.1 in [13], the proof of b) is given in [11]. The proof of a) follows along the lines of the proof of Theorem 4.1 in [13], the proof of b) is given in [11].…”

“…2 Lemma 2.2c) ensures for the stopping index N of the IRGNM the important estimate (see [12,13]) Using (2.3), the final residual satisfies r Jn X ≤ εγ n h Jn n X ≤ εCCγ n f (γ n ).…”

“…In [13] a complete convergence analysis of an inexact IRGNM with Morozov's discrepancy principle [17] as stopping criterion has been performed. In this article we analyze the complexity of this method.…”

A simplified generalized Gauss-Newton method for nonlinear ill-posed problems

“…Recently, Langer and Hohage [8] extended the analysis in [5] and [6] by considering (1.3) with the stopping rule (1.5) under a general source condition of the form…”