Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T00:47:57.704Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BEST WEIGHTED GRADIENT APPROXIMATION TO AN OBSERVED FUNCTION

Part of: Approximations and expansions General theory of linear operators

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 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. 

References

Aubin, J.-P., Applied Functional Analysis, 2nd edn (John Wiley, New York, 2000).Google Scholar
Howlett, P., ‘The best gradient approximation to an observed function’, Appl. Math. Sci. 2(28) (2008), 1365–1376.Google Scholar
Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, 88 (Academic Press, New York, 1980).Google Scholar
Luenberger, D. G., Optimization by Vector Space Methods (John Wiley, New York, 1969).Google Scholar
Naylor, A. W. and Sell, G. R., Linear Operator Theory in Engineering and Science, 2nd edn, Applied Mathematical Sciences, 40 (Springer, New York, 1982).CrossRefGoogle Scholar
Treves, F., Topological Vector Spaces, Distributions and Kernels (Academic Press, San Diego, 1967).Google Scholar
Yosida, K., Functional Analysis, 6th edn (Springer, Berlin, 1980).Google Scholar
Ziedler, E., Applied Functional Analysis, Applications to Mathematical Physics, Applied Mathematical Sciences, 108 (Springer, New York, 1997).Google Scholar