An iteratively regularized Gauss-Newton-Halley method for solving nonlinear ill-posed problems
This paper deals with the problem of the convergence of a Halley-type method for regularizing nonlinear inverse problems \(F(x)=y\) defined in Hilbert spaces. The method is based on the regularized Gauss-Newton iterative method but presents the novelty of including second derivatives of the involved operator \(F\). For a wide range of problems, for instance parameter identification problems for partial differential equations (PDEs), the operational cost of the evaluation of the second derivative is similar to the evaluation of the first derivative. This fact justifies the inclusion of second derivatives in the method. The author proves convergence and convergence rates in terms of the derivatives \(F'\) and \(F''\) depending on the regularity of the exact solution. The method can be applied in the ill-posed situation, where a noise level is assumed. The paper finishes with some numerical applications to the problem of identifying coefficients in some particular PDEs.
scientific article; zbMATH DE number 6481913
scientific article