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I'm trying to understand the underlying logic in the Radial Basis Function interpolation.

I understood that we estimate the value of the underlying function in any unknown point as

$ y(\vec{x}) = \sum \lambda_j \phi(r_j) $

where $r_j$ is the distance between the query point and $\vec{x}$.

In my understanding, we want the value of $y(\vec{x})$ to be "more similar" to the value of the underlying function in the points closer to $\vec{x}$. For this reason it makes sense to me to use basis functions as the inverse multiquadric or the gaussian that tends to zero as $r$ tends to infinity so that the "influence" of a point very far from my query point tends to zero.

However, reading the literature, I noticed that the most common and "original" basis function is the multiquadric one, defined as

$ \phi(r) = \sqrt{\left(r/\varepsilon\right)^2 + 1} $

that tends to $\infty$ as $r \rightarrow \infty $.

Could you please help me understanding why some basis function tends to zero, while other tends to infinity as $r \rightarrow \infty $, and still are both equally acceptable?

Luca
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  • The shape parameter is selected based on the size of your domain. Your intuition is right. MQ radial basis functions produce ill conditioned matrices because of the blow up nature of the RBF. But numerically the solutions to these matrices yield accurate results: http://www.scottsarra.org/math/papers/mqMonographSarraKansa.pdf – player100 Aug 07 '19 at 10:05
  • Does this mean that basis functions that tend to zero as r tends to infinity should be preferred? – Luca Aug 07 '19 at 12:36
  • No. As evidenced by the utility of MQs. But RBFs have many uses besides. If you are looking for an interpolant sometimes it makes sense to use RBFs with more local support – player100 Aug 07 '19 at 12:47

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