Stable gradient projection method for nonlinear conditionally well-posed inverse problems

Abstract

Abstract We study the standard gradient projection method in a Hilbert space, as applied to minimization of the residual functional for nonlinear operator equations with differentiable operators. The functional is minimized over a closed, convex and bounded set, which contains a solution to the equation. It is assumed that the inverse problem associated with the operator equation is conditionally well-posed with a Hölder-type modulus of relative continuity. We prove that the iterative process is asymptotically stable with respect to errors in the right part of the operator equation. Moreover, the process delivers in the limit an order optimal approximation to the desired solution.