A generalization of continuous regularized Gauss–Newton method for ill-posed problems

Abstract

Abstract A generalization of a simplified form of the continuous regularized Gauss–Newton method has been considered for obtaining stable approximate solutions for ill-posed operator equations of the form F(x) = y, where F is a nonlinear operator defined on a subset of a Hilbert space ℋ1 with values in another Hilbert space ℋ2. Convergence of the method for exact data is proved without assuming any specific source condition on the unknown solution. For the case of noisy data, order optimal error estimates based on an a posteriori as well as an a priori stopping rule are derived under a general source condition which includes the classical source conditions such as the Hölder-type and logarithmic type, and certain nonlinearity assumptions on the operator F.