Iterative Algorithms for Integral Equations of the First Kind,
Abstract
Integral equations of the first kind are usually ill-posed, that is, they have solutions which do not depend continuously on the right hand side. When solving these equations numerically, roundoff error is introduced in the right hand side, and even this small change can cause very large changes in the solution of the numerical problem. This problem is made even worse when the right hand side is observed with error, i.e. for ill-posed inverse problems. It is the purpose of this paper to do two things. First, we point out that, for a certain class of problems, simple Richardson iteration can provide a numerically stable means of approximately solving an Richardson's integral equation of the first kind numerically. However, Richardson's algorithm can converge very slowly. We therefore also discuss a preconditioned Richardson algorithm, which can greatly accelerate convergence and which has a natural probabilistic interpretation when applied to equations with positive, bounded kernels.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1992
- Accession Number
- ADP006629
Entities
People
- Mark G. Vangel