Numerische Mathematik

, Volume 111, Issue 4, pp 509–558

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

Article

DOI: 10.1007/s00211-008-0198-y

Cite this article as:
Jin, Q. & Tautenhahn, U. Numer. Math. (2009) 111: 509. doi:10.1007/s00211-008-0198-y

Abstract

We consider the computation of stable approximations to the exact solution \({x^\dagger}\) of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : XY between two Hilbert spaces X and Y by the Newton type methods
$$x_{k+1}^{\delta}=x_{0}-g_{\alpha_{k}}\left(F'(x_{k}^{\delta})^*F'(x_{k}^{\delta})\right) F'(x_{k}^{\delta})^*\left(F(x_{k}^{\delta})-y^{\delta}-F'(x_{k}^{\delta})(x_{k}^{\delta}-x_{0})\right)$$
in the case that only available data is a noise \({y^\delta}\) of y satisfying \({\|y^\delta - y\| \le \delta}\) with a given small noise level \({\delta > 0}\) . We terminate the iteration by the discrepancy principle in which the stopping index \({k_\delta}\) is determined as the first integer such that
$$\|F(x_{k_\delta}^{\delta})-y^{\delta}\|\le \tau \delta < \|F(x_{k}^{\delta})-y^{\delta}\|, \quad 0\le k < k_{\delta}$$
with a given number τ > 1. Under certain conditions on {αk}, {gα} and F, we prove that \({x_{k_\delta}^{\delta}}\) converges to \({x^\dagger}\) as \({\delta \rightarrow 0}\) 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 \({x_{0} - x^\dagger}\) is smooth enough.

Mathematics Subject Classification (2000)

65J15 65J20 47H17 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA
  2. 2.Department of MathematicsUniversity of Applied Sciences Zittau/GörlitzZittauGermany