Renormalised Steepest Descent in Hilbert Space Converges to a Two-Point Attractor

The result that for quadratic functions the classical steepest descent algorithm in Rd converges locally to a two-point attractor was proved by Akaike. In this paper this result is proved for bounded quadratic operators in Hilbert space. The asymptotic rate of convergence is shown to depend on the s...

Full description

Saved in:
Bibliographic Details
Published in:Acta applicandae mathematicae 2001-05, Vol.67 (1), p.1
Main Authors: Pronzato, Luc, Wynn, Henry P, Zhigljavsky, Anatoly A
Format: Article
Language:English
Subjects:
Algorithms
Eigenvalues
Eigenvectors
Hilbert space
Mathematical models
Partial differential equations
Studies
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The result that for quadratic functions the classical steepest descent algorithm in Rd converges locally to a two-point attractor was proved by Akaike. In this paper this result is proved for bounded quadratic operators in Hilbert space. The asymptotic rate of convergence is shown to depend on the starting point while, as expected, confirming the Kantorovich bounds. The introduction of a relaxation coefficient in the steepest-descent algorithm completely changes its behaviour, which may become chaotic. Different attractors are presented. We show that relaxation allows a significantly improved rate of convergence. [PUBLICATION ABSTRACT]
ISSN:0167-8019
1572-9036
DOI:10.1023/A:1010680020662