Smoothness for Some Selected Test Functions Relative to Shape Parameter via IMQ
radial basis functions, inverse multiquadric (IMQ), test function, shape parameter ()
Abstract
Radial basis function (RBF) approximation has the potential to provide accurate function approximations for large data site given at scattered node locations which yields smooth solutions for a given number of node points especially when the basis functions are scaled to be nearly at and when the shape parameter is choose wisely. In this paper, we concentrate on the choice of shape parameter, which must be choose wisely and the simplest strategy we adopt is to perform a series of interpolation experiments by varying the interval of shape parameter, and then pick the best" one. The best" was pick by checking the errors for different data sites and the smoothness of the error graphs. The results shows that the choice of interval for the shape parameter give better accuracy and smoothness of the graphs.
Downloads
How to Cite
References
J Biazar, M Hosami Selection of an interval for variable shape parameter in approximation by radial basis funtions.
Mehdi Dehghan, Ali Shokri (2009) Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. 230(2), 400-410.
J Wertz, E Kansa, L Ling (2006) The role of the multiquadric shape parameters in solving elliptic partial differential equations. 51(8), 1335-1348.
Ulrika Pettersson, Elisabeth Larsson, Gunnar Marcusson, Jonas Persson (2008) Improved radial basis function methods for multi-dimensional option pricing. 222(1), 82-93.
Mira Bozzini, Licia Lenarduzzi, Robert Schaback (2002) Adaptive Interpolation by Scaled Multiquadrics. 16(4), 375-387.
H Wendland (2005) Scattered data approximation.
S Deparis, D Forti, A Quarteroni (2014) A rescaled localized radial basis function interpolation on non-cartesian and nonconforming grids. 36(6), 2745-A2762.
E Larsson, E Lehto, A Heryudono, B Fornberg (2013) Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. 35(4), A2096A-2119.
M Buhmann (2003) Radial Basis Functions: Theory and Implementations. 12.
Gregory Fasshauer (2007) Meshfree Approximation Methods with Matlab.
Armin Iske (2002) Scattered Data Modelling Using Radial Basis Functions. 205-242.
G Fasshauer (1997) Solving partial di_erential equations by collocation with radial basis functions in Surface Fitting and Multire solution Methods. 131138.
A Fedoseyev, M Friedman, E Kansa (2002) Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary. 43(3-5), 439-455.
S Sarra, E Kansa (2009) Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations. 2.
Mehdi Tatari, Mehdi Dehghan (2009) On the solution of the non-local parabolic partial differential equations via radial basis functions. 33(3), 1729-1738.
E Kansa, R Carlson (1992) Improved accuracy of multiquadric interpolation using variable shape parameters. 24(12), 99120.
B Fornberg, G Wright (2004) Stable computation of multiquadric interpolants for all values of the shape parameter. 48, 853867.
M Bozzini, L Lenarduzzi, R Schaback (2002) Adaptive interpolation by scaled multiquadrics. 16, 375-387.
E Kansa, R Carlson (1992) Improved accuracy of multiquadric interpolation using variable shape parameters. 24(12), 99-120.
Bengt Fornberg, Julia Zuev (2007) The Runge phenomenon and spatially variable shape parameters in RBF interpolation. 54(3), 379-398.
Robert Schaback, Holger Wendland (2000) Adaptive greedy techniques for approximate solution of large RBF systems. 24(3), 239-254.
Song Xiang, Ke-Ming Wang, Yan-Ting Ai, Yun-Dong Sha, Hong Shi (2012) Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation. 36(5), 1931-1938.
S Sarra, D Sturgill (2009) A random variable shape parameter strategy for radial basis function approximation methods. 33(11), 12391245.
Published
2017-03-24
Issue
Section
License
Copyright (c) 2017 Authors and Global Journals Private Limited
This work is licensed under a Creative Commons Attribution 4.0 International License.