Are there some scattered point configurations that would yield bad interpolation results using Radial Basis Function (RBF) interpolation?

Question

Is Radial Basis Function interpolation sensible to the scattered point configuration?

I seem to be having problems for scattered points $(x_i,y_i)$ that are illustrated below:

The values $f(x,y)$ can be generated using an fBm generator.

Interpolation is then done using the RBF technique:

$ f(\textbf{x})=\sum_{k=1}^N c_k \phi(\lVert \textbf{x}-\textbf{x}_k\rVert) $

where $\textbf{x}_k$ are the scattered points illustrated above and $\textbf{x}=(x,y)$.

Answer 1

(Theoretically, for the interpolation system to be uniquely solvable, the points must be unisolvent, that is not to lie on a certain low-order polynomial specific to the chosen RBF: for example one cannot fit uniquely a bilinear polynomial to 3 points if they are collinear).

Much more relevant in practice is the ill-conditioning of RBF interpolation, and, indeed, its numerical sensitivity to the data and to the shape parameter of some RBF. Please search for more exploration on ill-condition RBF and pre-conditing methods. Practically, I'd suggest trying different RBF kernels, maybe even compactly supported RBF, as you don't seem to have irregularly large holes in your data and could supply a suitable support radius.