I would like to initiate a discussion on the role of shape parameter in RBF interpolation and applications. By shape parameter, I mean the one in new convention $\epsilon$ as suggested by Prof. G E Fasshauer. We all know that keeping the shape parameter small creates the problem of ill-conditioning and several works have been done to resolve this. My question is then, why do people recommend small shape parameters ? I have seen good accuracy in 2D RBF interpolation when I use large shape parameter with Gaussian RBF. Please provide your opinion on related aspects of the role of the shape parameter in this context. I have read the literature, but I want to know the views of people working with RBFs.
You can consider a question first: for a given function and a fixed point set, what is the best shape parameter in the sense of your error norm? then you can extend the question to a certain class of functions. I believe you will find the answer if you think about it carefully.
You can consider a question first: for a given function and a fixed point set, what is the best shape parameter in the sense of your error norm? then you can extend the question to a certain class of functions. I believe you will find the answer if you think about it carefully.
Usually the shape parameter is inversely proportional to the average distance between the data points, where "average" can be computed or estimated in many different ways. (There's considerable freedom.) A "small" shape parameter then is typical for sparse data. To deal with coarse sampling and a low-level information content, the RBF's must be relatively flat. However, I suppose that your question refers to why one should use a "relatively small" shape parameter. The reason, I guess, is to offer a little bit extra smoothness, as a too large shape parameter won't work -- the setting is more or less equivalent to the problem of "oversampling" in signal processing. Trying to fit basis functions (polynomials, trigonometric functions, RBF's) to non-existing information by increasing the degree, frequency (bandwidth), or using steeper shapes doesn't make sense. Thus it is better to err on the low side. Balance may be delicate, and the "best" parameter is only defined in terms of clearly stated and motivated objectives that also incorporate the limitations of the technique employed and the quality of the available data.
This follows from approximation theoretic principles. Essentially, this is problem dependent, and also depends on how you define the shape parameter. The question is not just of accuracy, but convergence, the rate at which the error drops off with increasing number of nodes. To get optimal convergence rates with global RBFs on smooth target functions, it has been determined that small shape parameters are necessary, because smooth shape parameters result in a more "global" basis; this allows us to recover spectral convergence. Depending on the problem you're solving, too large a shape parameter may result in unphysical behavior. Greg's book has an example of this, where an RBF-PS solution transitions at a different time than a chebyshev spectral method.
In practice, as you said, ill-conditioning can offset this. Further, larger shape parameters may be more appropriate for target functions that aren't very smooth; if you're recovering a C2 function with infinitely-smooth RBFs, a small shape parameter will most certainly not be necessary.
There are several modern algorithms that can completely bypass the ill-conditioning issue. The current best is the RBF-GA algorithm, limited to Gaussian RBFs. This is a variant on the RBF-QR technique that Bengt Fornberg and Cecile Piret came up with. There are other modern variants: the Hilbert-Schmidt SVD approach that Greg and Mike McCourt came up with can be shown to be equivalent to RBF-QR, but approaches the problem from the perspective of Mercer's theorem and eigenfunction expansions. Stefano DeMarchi also has what he calls the Weighted SVD method, which works with any RBF basis function, but requires a quadrature/cubature rule, and only partially offsets ill-conditioning. There are also a couple of other fascinating algorithms: the Contour Pade algorithm, which treats the RBF interpolant as a meromorphic function and uses contour integration to bypass any pole singularities, and its modern (unpublished) variant, the RBF-RA approach, which replaces the Pade approximation techniques in the Contour-Pade method with a Rational approximant.
The small shape parameter limit is also quite important when you view the connections of RBFs to polynomials, Fourier series and spherical harmonics. In 1D, on Chebshev nodes, as the shape parameter goes to zero, you recover the polynomial interpolant (in an analytic sense, and using the new stable algorithms). On the circle, as the shape parameter goes to zero, spherical RBFs recover the Fourier interpolant. On the sphere, they recover Spherical Harmonics. In 2D and higher, if the node set is polynomial unisolvent, the RBF interpolant will recover that polynomial in this limit.
I can add that there are theoretical results where the errors depend on the shape parameter.
I can add that : It all depends on the chosen RBF, you can see my paper.
Best regards
Conference Paper Numerical experiments on the performance of the RBF meshfree...
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