The flat RBF limit
Motivation
As RBFs become increasingly flat (with our notation, as the shape parameter goes to zero), a typical interpolation matrix becomes more and more ill-conditioned. This is not surprising as all basis function in the limit approach the constant function and the limit matrix of all ones has rank one.
However, more surprising is that the limit itself, if it can be computed by some other approach than the direct one, is well defined and reproduces multivariate polynomial interpolation.
Consequences of this are:
- Interpolation with nearly flat RBFs give very accurate results for smooth functions (well approximated by polynomials).
- If node points are chosen for example as Chebyshev points, the flat RBF limit reproduces pseudo-spectral Chebyshev methods. Furthermore, RBF interpolation in the limit can lead to spectral accuracy also for scattered nodes in non-trivial geometries.
The figure shows the error as a function of the shape parameter when interpolating a rational function over the unit disc using the direct solution approach (dashed red) and the RBF-QR method (solid black). In each case, the result using double precision and quad precision arithmetic is shown. The red curves are completely corrupted for small shape parameters due to ill-conditioning, whereas the black ones are correct almost to machine precision. N=402 radially clustered node points were used.
Contributions
- Understanding the flat multivariate RBF limit
The first paper here explores the behaviour of RBF limits for special node configurations and show that for standard RBFs such as inverse quadratics and multiquadrics, the limit may not exist. In the second paper, we show how the multivariate limit behaviour is related to polynomial unisolvency and prove under which conditions the limit is the unique polynomial interpolant, a polynomial interpolant that depends on the RBF, or diverges. The third paper is about the special case when all nodes lie on a circle. Normally, this prevents divergence as showed by Buhmann and Dinew in an earlier paper, but if the Taylor series of the RBF has vanishing coefficients this changes.- A note on radial basis function interpolant limits. In IMA Journal of Numerical Analysis, volume 30, pp 543-554, 2010. (DOI).
- Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. In Computers and Mathematics with Applications, volume 49, pp 103-130, 2005. (DOI).
- Some observations regarding interpolants in the limit of flat radial basis functions. In Computers and Mathematics with Applications, volume 47, pp 37-55, 2004. (DOI).
- Basis functions with special limit properties
Gaussian RBFs never diverge in the limit. This was conjectured in one of our papers and proved by Robert Schaback (Constr. Approx. 21 (2005)). In the paper below, we show that Gaussians is the extreme case in a class of Bessel RBFs, which all have special limit properties.- A new class of oscillatory radial basis functions. In Computers and Mathematics with Applications, volume 51, pp 1209-1222, 2006. (DOI).
- Computing interpolants and approximants in the flat limit
The first approach to compute interpolants in the flat RBF limit was constructed by Fornberg and Wright (Comput. Math. Appl. 48 (2004)). This approach is limited to small to intermediate problem sizes. Therefore, we have been looking at other approaches and now have the RBF-QR method, which works in up to three space dimensions right now. The technical report below only covers the 2-D case, but a manuscript covering also the 1-D and 3-D cases has been accepted for publication in SIAM J. Sci. Comp. See also the RBF-QR method on the sphere by Fornberg and Piret (SIAM J. Sci. Comp. 30 (2007)).- Stable computations with Gaussian radial basis functions in 2-D. Technical report / Department of Information Technology, Uppsala University nr 2009-020, 2009. (External link).
Ongoing research
We are continuing our investigations of the effects of scaling and node placement on the accuracy and stability of global and local RBF approximations.