Iteratively Weighted Approximation Algorithms For Nonlinear Problems Using Radial Basis Function Examples

Abstract

A set of discrete data (x_k, f (x_k)) (k = 1, 2, …, m) may be fitted in any l_p 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 l_p 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 ϵ ≡ f – g (L) orthogonal to a set of functions forming a basis for L, and the second based on making the implicit error ϵ* ≡ 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)