Numerische Mathematik

, Volume 123, Issue 4, pp 745-779

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

  • Thorsten HohageAffiliated withInstitute for Numerical and Applied Mathematics, University of Göttingen
  • , Frank WernerAffiliated withInstitute for Numerical and Applied Mathematics, University of Göttingen Email author 


We study Newton type methods for inverse problems described by nonlinear operator equations \(F(u)=g\) in Banach spaces where the Newton equations \(F^{\prime }(u_n;u_{n+1}-u_n) = g-F(u_n)\) are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss–Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to \(0\) both for an a priori stopping rule and for a Lepskiĭ-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback–Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performance of the proposed method for these problems is illustrated in numerical examples.

Mathematics Subject Classification (2000)

65J15 65J20 78A46 65K10