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Stable gradient projection method for nonlinear conditionally well-posed inverse problems. (English) Zbl 1338.65153

Summary: 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.

MSC:

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65J22 Numerical solution to inverse problems in abstract spaces
47J06 Nonlinear ill-posed problems
47J25 Iterative procedures involving nonlinear operators
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