Iterative Algorithms for Integral Equations of the First Kind With Applications to Statistics

Abstract

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 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.

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Document Details

Document Type
Technical Report
Publication Date
Oct 01, 1992
Accession Number
ADA256529

Entities

People

  • Mark Vangel

Organizations

  • Harvard University

Tags

Communities of Interest

  • Energy and Power Technologies
  • Human Systems

DTIC Thesaurus Topics

  • Algorithms
  • Asymptotic Series
  • Composite Materials
  • Computational Fluid Dynamics
  • Computational Science
  • Data Science
  • Differential Equations
  • Equations
  • Functional Analysis
  • Information Science
  • Integral Equations
  • Inverse Problems
  • Materials Science
  • Numerical Analysis
  • Probability
  • Random Variables
  • Theorems

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Regression Analysis.