Improved accuracy of multiquadric interpolation using variable shape parameters

Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

“…There are different approaches to distribute the values of the variable shape parameter over an interval. Kansa and Carlson presented some variable shape parameter strategies with different shape parameters at different centres [16]. Their distributions of parameters are in the forms of increasing linearly, decreasing linearly, and exponentially varying shape parameter, respectively, as follows:…”

“…Sarra and Sturgill emphasize the importance of the length of the interval, = max − min , they take = 1, and they study the errors by trial and errors, while max and min vary in a meaningful range [19]. Kansa and Carlson [16] consider a set of shape parameters which minimizes an error function over some evaluation points. To find these values, the following objective function is minimized:…”

“…Advances in Numerical Analysis 3 They had shown that variable form of the MQ improves the accuracy of the interpolant in the interior regions where the fitted function varies relatively rapidly [16]. Sarra and Sturgill introduced a random variable shape parameter strategy in interpolations and two-dimensional linear elliptic boundary value problems as follows [19]:…”

“…It is acknowledged that in many cases using variable shape parameter strategies results in more accurate approximation [16,17,19]. A variable shape parameter strategy uses different values of the shape parameter at different centre points:…”

“…Several articles have been dedicated to introducing different algorithms to compute a constant value, as an optimal or a good value, for the shape parameter [6][7][8][9][10][11][12][13][14][15]. Kansa and Carlson showed that instead of a fixed shape parameter variable shape parameters are also useful [16]. They distribute some values in an interval to use them in variable shape parameter.…”