Some approximate Gauss-Newton-type methods for nonlinear ill-posed problems (Q2871314)

The paper deals with the numerical solution of a nonlinear equation \(F(x) = 0\) in Hilbert spaces. \(F\) is supposed to be Fréchet differentiable. It is possible that the problem is ill-posed. A (generalized) solution is to find in the least squares sense by minimizing the functional \([\|F(x)\|]^2\). The stationary points are solutions of the normal equation. The proposed iterative scheme is a procedure based on the Tikhonov functional. It is a class of two-parameter regularized Gauss-Newton-type methods. The generated sequence converges to the solution of \(F(x) = 0\) under some assumptions on the regularization parameters, the relaxation parameters and the test function, respectively. Error estimations are derived. In order to illustrate the accuracy of the method, two sample problems are studied. The first one is a linear-quadratic algebraic system in four variables. The second one is an integral equation which is equivalent to a Volterra integral equation of first kind. The analysis of these test examples is quite complicated. Therefore, the proposed method does not seem to be an effective tool for finding solutions.