Iterative Algorithms for Integral Equations of the First Kind with Applicationsto Statistics

摘要

This dissertation explores the use of a preconditioned Richardson iterativealgorithm 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 Richardson's 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.