Theory and Methods Related to the Singular-Function Expansion and Landweber's Iteration for Integral Equations of the First Kind

For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^\ast(g - K \hat{f}_{k - 1}),\quad k = 1, 2, \cdots,$ both with respect to its convergence properties and its response to sin...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 11; no. 4; pp. 798 - 825
Main Author: Strand, Otto Neall
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.09.1974
Subjects:
Ambiguity
Approximation
Differential equations
Eigenfunctions
Error function
Error rates
Integral equations
Iterative solutions
Mathematics
Methods
Perceptron convergence procedure
Polynomials
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^\ast(g - K \hat{f}_{k - 1}),\quad k = 1, 2, \cdots,$ both with respect to its convergence properties and its response to singular functions of K. Here K* is the adjoint of K, f̂0 is a suitable starting function and D is a fixed linear operator to be chosen. The general theory of singular functions is interpreted and extended for the study of the iteration, and a quantitative method of choosing D to shape the response to singular functions of K is derived. Several specific matrix and integral-equation examples are presented.
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ISSN:0036-1429
1095-7170
DOI:10.1137/0711066