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We study the standard gradient projection method in a Hilbert space, as applied to minimization of the residual functional for nonlinear operator equations with di erentiable 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.

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We present a class of iterative methods for solving nonlinear ill-posed operator equations with differentiable operators in a Banach space. The methods are based on regularization of the linearized equation by using an appropriate regularization scheme at each iteration. We establish that the iterations converge locally with power rate provided that the solution admits of a sourcewise representation. We also prove that the condition of sourcewise representability is very close to a necessary condition for this kind of estimates.

No abstract