The mystery of the shape parameter IV

Abstract

This is the fourth paper of our study of the shape parameter c contained in the famous multiquadrics (−1) ⌈β⌉ (c 2 + x 2 ) β , β > 0, and the inverse multiquadrics (c 2 + x 2 ) β , β < 0. The theoretical ground is the same as that of [10]. However we extend the space of interpolated functions to a more general one. This leads to a totally different set of criteria of choosing c.

The mystery of the shape parameter III

“…It is clearly seen that in Tables 1-6, the interpolation error RM S is always very small when c is equal to the optimal value c 0 suggested by the MN curves, even if this choice may not be the experimentally best one. In our previous papers Luh [9,10,11], one can always find the experimentally best c according to the MN curves by decreasing the value of δ. However, here we scrapped this approach for two reasons.…”

The shape parameter in the shifted surface spline III

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The purpose of this article is to explore the optimal choice of shape parameter which is an important and longstanding problem in the theory of radial basis functions(RBF). We already handled it for multiquadric and Gaussian in [9,10,11,12,13]. Here we focus on shifted surface spline and present concrete criteria for the choice of shape parameter.

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A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation

“…For any f ∈ E σ and x ∈ Ω, the upper bound of |f (x) − û(x)| is a very complicated expression involving both ρ and ∆ 0 , as can be seen in Luh [7]. A modified theory for purely scattered data setting can be seen in Luh [8].…”