Rocha [1] claims that there is an example of cubic RBF giving singular system in Powell's book. But I did not find it there. In chapter 3 (p.30-31) of Hubbert's book [2], they claim it happens but they only mention Thin Plate Spline ( not cubic RBF).
So the question is, has anybody come across any paper, where someone actually had shown a case of cubic RBF giving singular system? There is no proof that it is non-singular but I have never come across any example where it is singular for 2D interpolation.
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[1] Rocha, H., 2009. On the selection of the most adequate radial basis function. Applied Mathematical Modelling 33 (3), 1573 – 1583
[2] Hubbert, S., Lê Gia, Q. T., Morton, T. M., 2015. Spherical Radial Basis Functions, Theory and Applications. Springer International Publishing
@Ariadne - Thanks for sending "my own paper" to me.
Anyway,
After talking to some experienced people working with RBF, I finally found an explanation to this.
I found the singular system by taking point (0,0), (0,1), (1,0) and (x,y) and interpolating them using cubic spline RBF.
The determinant of interpolation matrix becomes zero for a number of points, lying on the zero level contour.
Please see the matlab code in the attachment.
An interesting question in this context concerns overcompleteness of the basis set. This is often related to (quasi) singular problems. For a discussion of convergence issues see also the attached link. It stems from using Gaussian RBF in quantumchemical calculations, but discusses some general ideas that may be helpful also for different fields of applications.
http://www.maths.warwick.ac.uk/research/2004_2005/symposium/kutzelnigg1.pdf
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