  Iteratively weighted approximation algorithms for nonlinear problems using radial basis function examples

Jenkinson, D.P., Mason, John C. and Crampton, Andrew (2004) Iteratively weighted approximation algorithms for nonlinear problems using radial basis function examples. Applied Numerical Analysis & Computational Mathematics, 1 (1). pp. 165-179. ISSN 1611-8170

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Abstract

A set of discrete data (xk, f (xk)) (k = 1, 2, , m) may be fitted in any lp norm by a nonlinear form derived from a function g (L) of a linear form L = L(x). Such a nonlinear approximation problem may under appropriate conditions be (asymptotically) replaced by the fitting of g-1 (f) by L in any lp norm with respect to a weight function w = g (g-1 (f)). In practice this direct method can yield very good results, sometimes coming close to a best approximation. However, to ensure a near-best approximation, by using an iterative procedure based on fitting L, two algorithms are proposed in the l2 norm - one already established by Mason and Upton (1989) and one completely new, based on minimising the two algorithms and multiplicative combinations of errors, respectively. For a general g we prove they converge locally and linearly with small constants. Moreover it is established that they converge to different (nonlinear) Galerkin type approximations, the first based on making the explicit error f - g (L) orthogonal to a set of functions forming a basis for L, and the second based on making the implicit error * w(g-1 (f) - L) orthogonal to such a basis. Finally, and mainly for comparison purposes, the well known Gauss-Newton algorithm is adopted for the determination of a best (nonlinear) approximation. Illustrative problems are tackled and numerical results show how effective all of the algorithms can be. To add a further novel feature, L is here chosen throughout to be a radial basis function (RBF), and, as far as we are aware, this is one of the first successful uses of a (nonlinear) function of an RBF as an approximation form in data fitting. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Item Type: Article Q Science > QA Mathematics > QA75 Electronic computers. Computer scienceQ Science > QA Mathematics School of Computing and EngineeringSchool of Computing and Engineering > Automotive Engineering Research GroupSchool of Computing and Engineering > High-Performance Intelligent ComputingSchool of Computing and Engineering > High-Performance Intelligent Computing > Planning, Autonomy and Representation of KnowledgeSchool of Computing and Engineering > High-Performance Intelligent Computing > Planning, Autonomy and Representation of Knowledge?? tserg ??School of Computing and Engineering > High-Performance Intelligent Computing > Information and Systems Engineering Group Briony Heyhoe 15 Sep 2008 13:22 28 Aug 2021 10:41 http://eprints.hud.ac.uk/id/eprint/1860 View Item