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Tikhonov regularization in Hilbert scales for nonlinear statistical inverse problems. (English) Zbl 1359.65004

Tönning: Der Andere Verlag: Göttingen: Univ. Göttingen, Mathematisch-Naturwissenschaftliche Fakultäten (Diss.) (ISBN 978-3-89959-684-7/pbk). 101 p. (2007).
Summary: This PhD thesis deals with regularization theory for deterministic and statistical inverse problems, with a special interest for nonlinear problems in a statistical framework. We considered the inverse problem to estimate a quantity , which is not directly observable but is related to an accessible vector with the help of a nonlinear, one-one operator which has a non-continuous inverse. The data which we have at our disposal is modelled by a general noise model and contains both deterministic and stochastic noise. We constructed an estimator with the help of Tikhonov regularization in Hilbert scales, method chosen to cope with the discontinuity of the inverse of the operator. We showed that this leads to order-optimal rates of convergence for the expected error for a range of smoothness classes, for a-priori choice of the regularization parameter and under some assumptions on the operator. We studied the general optimality of our method with respect to the stochastic noise level, and we concluded that the minimax rates are attained in the case of linear operators. We also discussed the optimality of our method for nonlinear operators.
Typically, a-priori rules require a-priori knowledge of the smoothness of the solution to obtain optimal rates of convergence whereas certain a-posteriori rules do not need such a-priori knowledge. Therefore, we studied Lepskii’s balancing principle for our estimator. We obtained order-optimal rates of convergence for this a-posteriori choice of the regularization parameter in the deterministic setting and almost-optimal in the stochastic setting. We considered two parameter identification problems for elliptic differential equations and a Hammerstein integral equation, and we showed that the rates of convergence proved before for our estimator are valid for these examples, since the assumptions of the theorems hold for these problems. This involved the construction of appropriate Hilbert scales for each example. We illustrated by Monte Carlo simulations the rates of convergence proved for a-priori choice of the regularization parameter for one of the parameter identification problem studied before. It turned out that the rates exhibited by our numerical estimator matched those predicted by the convergence theorems.

MSC:

65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
62-02 Research exposition (monographs, survey articles) pertaining to statistics
65C60 Computational problems in statistics (MSC2010)
65J15 Numerical solutions to equations with nonlinear operators
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

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