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THE BEST WEIGHTED GRADIENT APPROXIMATION TO AN OBSERVED FUNCTION

Part of: General theory of linear operators Approximations and expansions

Published online by Cambridge University Press:  14 October 2014

PHIL HOWLETT
PHIL HOWLETT*
Affiliation:
Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), School of Information Technology and Mathematical Sciences, University of South Australia, South Australia 5095, Australia email p.howlett@unisa.edu.au
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Abstract

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We find the potential function whose gradient best approximates an observed square integrable function on a bounded open set subject to prescribed weight factors. With an appropriate choice of topology, we show that the gradient operator is a bounded linear operator and that the desired potential function is obtained by solving a second-order, self-adjoint, linear, elliptic partial differential equation. The main result makes a precise analogy with a standard procedure for the best approximate solution of a system of linear algebraic equations. The use of bounded operators means that the definitive equation is expressed in terms of well-defined functions and that the error in a numerical solution can be calculated by direct substitution into this equation. The proposed method is illustrated with a hypothetical example.

Keywords

gradient operatorbest approximationSobolev spaces

MSC classification

Primary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A50: Equations and inequalities involving linear operators, with vector unknowns
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 54 - 68
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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