Improved accuracy of multiquadric interpolation using variable shape parameters

Given N scattered data points, we examined the problem of finding N variable Multiquadric (MQ) shape-parameters, or R² values. Because the problem of finding the optimal R² values is a nonlinear one, we optimized these parameters numerically by minimizing the root-mean-square (RMS) errors. The resulting R² values varied over many orders of magnitude. We have tested this approach on a number of univariate and bivariate (Franke's) problems, and found that the RMS error reduction was substantial.