On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

Qinian Jin*, Ulrich Tautenhahn

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

44 Citations (Scopus)

Abstract

We consider the computation of stable approximations to the exact solution x of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods xk+1δ=x0-g αk(F'(xkδ) *F'(xkδ) F'(xk δ)*(F(xkδ)-y δ-F'(xkδ)(xk δ-x0) in the case that only available data is a noise yδ of y satisfying ||yδ- y|| ≤ δ with a given small noise level δ > 0. We terminate the iteration by the discrepancy principle in which the stopping index k δ is determined as the first integer such that ||F(x δ)-yδ||≤ τ δ < ||F(xkδ)-yδ||, 0≤ k < kδ with a given number τ > 1. Under certain conditions on {α k }, {g α } and F, we prove that xδ converges to x as δ → 0 and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fréchet derivative F' of F if x0 - x is smooth enough.

Original languageEnglish
Pages (from-to)509-558
Number of pages50
JournalNumerische Mathematik
Volume111
Issue number4
DOIs
Publication statusPublished - Feb 2009
Externally publishedYes

Fingerprint

Dive into the research topics of 'On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems'. Together they form a unique fingerprint.

Cite this