Hi,

I'm trying to implement radial basis function (RBF) for interpolation on overset grid and i've some questions...

1. Is RBF valid for both local & global interpolation?

2. Is RBF interpolation suitable for structured grid?

3. I tried to let the shape parameter c=1.25D/sqrt(N) Where D is the diameter of the minimal circle enclosing all data points and N is the number of data points. At some locations, the inverse matrix becomes singular. Can I prevent that by changing the c?

Thanks alot!

You can use it for both local and global interpolation. But with local interpolation it will not be very accurate. The spectral character of RBF comes from its global nature.

The shape parameter has a very critical influence on the accuracy of RBF. There is no rational way of selecting this at present. There is an uncertainty principle between accuracy and condition number; both cannot be small at the same time.

Ill conditioning occurs when the basis functions become very flat, but an interpolant still exists, just that you cannot get it by inverting the matrix, see

Interpolation In The Limit Of Increasingly Flat Radial Basis Functions http://citeseer.ist.psu.edu/308829.html

You can compute the interpolant even in case of ill-conditioned matrix using special methods, see

B. Fornberg, G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl. 16 (2004) 497-523.

You will have to experiment with different values of the shape parameter. Or you can use a validation approach to "optimize c", see for example,

Shmuel Rippa. "An algorithm for selecting a good value for the parameter c in radial basis function interpolation." Adv. Comput. Math., 11(2-3):193--210, 1999.