Iterative Algorithms for Integral Equations of the First Kind With Applications to Statistics
HARVARD UNIV CAMBRIDGE MA DEPT OF STATISTICS
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This dissertation explores the use of a preconditioned Richardson iterative algorithm for the solution of linear and nonlinear ill-posed integral equations of the first kind. The discussion consists of three parts, which can be roughly categorized as numerical analysis, applications to statistical methodology, and an application to an inverse problem. In the first part, singular matrix equations that result from discretizing ill-posed integral equations of the first kind are considered. Sufficient conditions for the convergence of Richardsons algorithm to a solution are established, and necessary and sufficient conditions are proven for special cases. The inconsistent case is also discussed. A preconditioning for equations with positive kernels leads to the Conditional Expectation algorithm, which is discussed in detail. A notion of iterative regularization is introduced and related to the more usual penalized least squares approach to regularization.
- Numerical Mathematics