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Morozov’s discrepancy principle for Tikhonov-regularization of nonlinear operators. (English) Zbl 1002.65064
For a nonlinear operator \(F\) between Hilbert spaces assume that the solutions of the equation \(F(x) = y\) do not depend continuously on the data. Morosov’s principle concerns a posteriori selection strategies for the regularization parameter \(\alpha\) in the Tikhonov functional. Under certain restrictions on \(F\) and the solution, the author shows that parameters \(\alpha\) can be guaranteed to exist such that \(\delta \leq \|y^{\delta} - F(x^{\delta}_{\alpha} \|\leq c \delta\) holds and then derives a convergence rate result. Several examples illustrate that the required conditions are met for a large class of operators relevant in practice.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J06 Nonlinear ill-posed problems
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