CISL Annual Report 2005 > CISL research accomplishments

Numerical methods: Radial Basis Functions accomplishments

Radial basis functions (RBFs) is a new numerical methodology that offers extreme high-order accuracy with scattered nodes on completely irregular geometries. Moreover, it is algorithmically much simpler than other methods such as spectral elements. However, it is still in a developmental stage and much research is needed before it can be applied to large-scale production models. But the frontier looks exceptionally promising.

CSS continues its research in the area of radial basis functions. The primary areas of interest are to better understand the interpolation properties for oscillatory Bessel RBF and to extend RBF theory to spherical geometries to develop grid-free approaches for solving time-dependent PDEs on the sphere. Activities in 2005 include:

1. Working with researchers at the University of Utah, we demonstrated a successful application of RBFs to purely hyperbolic equations in spherical geometries. This was a first: the literature has no record of anyone stably time-stepping an RBF spatial discretization scheme for purely hyperbolic equations.

   To illustrate the outstanding performance of RBFs, a comparison was made with spherical harmonics (SH), double Fourier series (DF), and spectral elements on a cube (SE) using the test case of pure advection of a cosine bell directly over the poles. The following table summarizes the results of this comparison:

   Method	Cost per time step	L∞ Error N=4608	Code length (lines)	Local mesh refinement	Complexity increases with dimension
   RBF	O(N2)	0.003	< 50	Yes	No
   SH	O(N3/2)	0.04	> 500	No	Yes
   DF	O(N logN)	0.04	> 100	No	Yes
   SE	O(k N)	0.02	> 1000	Yes	Yes

2. With researchers in NCAR's newly formed Institute for Mathematics Applied to Geosciences (IMAGe) and at the Colorado School of Mines, the development and application of fast RBF interpolation algorithms for large climatological data sets from scattered stations (both in time and space) is being developed. The result will allow for the interpolant to the data to be calculated in O(N) operations as opposed to in O(N²) operations with classically employed methods.

3. A research group on RBFs was formed at the University of Colorado at Boulder, composed of four female Ph.D. students in Applied Mathematics. Recently, a paper has been submitted with Dr. Bengt Fornberg and two of the Ph.D. students as co-authors on the localization properties of the coefficients in RBF expansions. Such basic research will lead to the development of faster algorithms for the computation of the interpolant to a data set.