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

This paper treats numerical methods for solving the nonlinear ill-posed equation F(x) = 0, where the operator F is a Fr´ echet differentiable operator from one Hilbert space into another Hilbert space. Two parametric approximate Gauss-Newton- type methods are developed, a local convergence theorem is proved under certain conditions on a test function and the required solution, and some computational aspects are discussed. The validity of the theoretical convergence rate estimates is illustrated by the numerical results of solving two sample problems, one in a finite-dimensional and the other in an infinite-dimensional Hilbert space.