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Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems with Noisy Data.

Abstract : We discuss a number of topics related to the practical solution of ill posed problems given noisy data as it might arise in an experimental situation. The sensitivity of a regularized estimate of f to the noise is made explicit. After giving the intrinsic rank of the examples of first and second derivative, Abel's equation and Fujita's equation, it is argued that the first three are only mildly ill posed and f should be amendable to accurate estimation by the method of regularization. The method of Generalized Cross Validation (GCV) for choosing the regularization parameter is described and numerical results for the estimation of first and second derivative from noisy data are given. Two numerical algorithms for obtaining a regularized estimate with GCV are detailed. The second uses a B-spline basis to allow the handling of large data sets. This use of outside information in the estimation of f is discussed. We discuss the problem of checking the validity of the 'model' k, and provide a crude goodness-of-fit test. Finally we end by describing the (known) result that the number k of iterations in a Landweber iteration for solving large linear systems is a form of regularization parameter. We than show how GCV can feasibly be used to choose k in very large problems like those arising in computerized tomography.