Local convergence analysis of the Gauss-Newton-Kurchatov method
Ioannis K. Argyros, Stepan Shakhno
TL;DR
The paper develops a local convergence theory for the Gauss-Newton-Kurchatov method applied to nonlinear least squares with a decomposed residual $F(x)+G(x)$, where $F$ is differentiable and $G$ may be nondifferentiable. By formulating the iteration with $A_n=F'(x_n)+G(2x_n-x_{n-1},x_{n-1})$ and using divided differences for $G$, it derives convergence under center-Lipschitz and restricted/special Lipschitz conditions, introducing a refined convergence radius $r_*$ and explicit error bounds. The main result shows the iterates remain in a convergence ball $oldsymbol{igl( ext{x}^*,r_*igr)}$ and converge to the solution $x^*$, with quadratic convergence in the zero-residual case and linear convergence for small residuals, provided certain norm bounds hold. Numerical experiments demonstrate faster convergence of the combined differential-difference schemes compared with Kurchatov- and Secant-type methods, validating the theoretical improvements and practical advantage of avoiding full Jacobian computations for non-differentiable components.
Abstract
We present a local convergence analysis of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems with a decomposition of the operator. The method uses the sum of the derivative of the differentiable part of the operator and the divided difference of the nondifferentiable part instead of computing the full Jacobian. A theorem, which establishes the conditions of convergence, radius and the convergence order of the proposed method, is proved (Shakhno 2017). However, the radius of convergence is small in general limiting the choice of initial points. Using tighter estimates on the distances, under weaker hypotheses (Argyros et al. 2013), we provide an analysis of the Gauss-Newton-Kurchatov method with the following advantages over the corresponding results (Shakhno 2017): extended convergence region; finer error distances, and an at least as precise information on the location of the solution. The numerical examples illustrate the theoretical results.