## Abstract

We consider the methods x_{n+1}^{δ} = x _{n}^{δ}-g_{αn} (F′(x_{n} ^{δ})*F′( x_{n}^{δ}))F′( x_{n}^{δ})′(F(x_{n}^{δ})- y^{δ}) for solving nonlinear ill-posed inverse problems F(x) = y using the only available noise data y^{δ} satisfying ∥y ^{δ} - y∥ ≤ δ with a given small noise level δ >0. We terminate the iteration by the discrepancyprinciple ∥F(x _{nδ}^{δ})-y^{δ} ∥ ≤ τ^{δ} ≤ ∥F(x_{n}^{δ})-y ^{δ} ∥, 0 ≤ n < n_{δ}, with a given number τ > 1. Under certainconditions on {α_{n}} and F, we prove for a large class of spectral filter functions {gα} the convergenceof x_{nδ}^{δ} to a true solution as δ → 0. Moreover, we derive the order optimal rates of convergence when certain Hölder source conditions hold. Numerical examples are given to test the theoretical results.

Original language | English |
---|---|

Pages (from-to) | 549-573 |

Number of pages | 25 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 49 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |