THE BEST WEIGHTED GRADIENT APPROXIMATION TO AN OBSERVED FUNCTION

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 fu...

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Bibliographic Details
Published inJournal of the Australian Mathematical Society (2001) Vol. 98; no. 1; pp. 54 - 68
Main Author HOWLETT, PHIL
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.02.2015
Subjects
primary 41A65
Sobolev spaces
best approximation
secondary 47A05
47A50
gradient operator
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ISSN1446-7887
1446-8107
1446-8107
DOI10.1017/S1446788713000621

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Summary: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.
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ISSN:1446-7887
1446-8107
1446-8107
DOI:10.1017/S1446788713000621