Dear all,
The appearance of stable algorithms (such as RBF-QR, RBF-GA and others) show that the RBF interpolation operator should be uniformly bounded by the data. Stefano & Robert proved this for finitely smooth basis functions with Fourier transforms decaying only algebraically (Adv. Comput. Math. 2010). However stable algorithms work for Gaussian and MQ as well. Does there exist a mathematical proof for stability of interpolation operator of infinitely smooth basis functions?
Thanks.
I've looked into this, and the closest I can find is a more indirect argument. In 1D, Rodrigo Platte showed that the Gaussian is just a mapped polynomial. In addition, Fornberg and co-workers showed that the flat limit is a polynomial. If the flat limit is a polynomial, it follows that RBFs in the flat limit will have the same stability as the unique polynomial interpolant to the data. In other words, the RBF Lebesgue constants must converge to the polynomial Lebesgue constant. I think this is the closest I've seen. In greater than 1D, there's Schaback's proof that scaled Gaussians converge to the least interpolant of de Boor and Ron, implying again that the Lebesgue constants of the former converge to those of the latter.
I'm sorry I can't be more specific. You seem to have found an interesting gap in the literature. Or maybe I'm missing something.
Personal opinion: the stability of RBF depends on the computer accuracy. The last few decimal number of weights may cause the instability. When uniform nodes are used, few local number of nodes lead to the same weights, and then the stability enhanced. Prof. Fornberg has done plenty work to mention this issue from the engineering point of view. And Prof. Schaback is of course one the pioneers of "Theory proof". It is not easy to proof it. We are looking forward to see what will happen!
An update: Robert Schaback just posted this article on Researchgate showing the stability of kernel-based interpolation for Gaussians and Sobolev kernels. See here: Article Stability of radial basis function interpolants
Varun Shankar
Dear Varun,
This paper proves the stability for finitely smooth kernels and thus excludes Gaussian. It uses a bump function argument in Sobolev spaces. But bump functions may not exist in native space of Gaussian [Robert].
Bests.
Huh. I stand corrected. Was that a quote from Schaback himself? Regardless, I scanned through the paper I myself posted, and you're right, it only proves stability for finitely-smooth kernels.
In 1D, we know that Gaussian RBF interpolation with shape parameter going to zero is unstable on equispaced points since it recovers the Lagrange interpolating polynomial. The Lebesgue constant must grow without bound here. In 2D and higher, it recovers de Boor and Ron's "least interpolant", a unique polynomial interpolant on scattered points. Again, this indicates that for appropriate point sets, Gaussian RBFs must be unstable in arbitrary dimension (with uncontrolled growth of Lebesgue constants).
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