RBF to build a shape function, does it really make a compact support shape function ?

Hi





I am trying to build a shape function shape function using compatly supported RBF.

The system i am solving is:

Kij B  *   λ  = I

B^T 0      μ     0

Where K is the RBF,  Kij - Kji = K (|| xi - xj  ||) , B is the base [1 x y]

I is identity and allow me to have dij shape function (1 0 0 0) then ( 0 1 0 0) etc



I evaluate the interpolator as folow



Sum( λi( K(|| x - xi||)) ) + Sum ( μl B(x) )



It is my understanding that by solbing such as system of equation we first fit the base to the data using least square



Then add the missing bit to make an interpolating function using RBF



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I'm trying to use those shape function as a local interpolation.

However i do not see how those shape function have a compact support at all.





Example using quartic spline as K, and eucledian norm i have: (see attachment)

I can see there's a limited domain on which K influence the plane to have dij.

But considering that this shape function is the effect on the whole interpolation of the value of the node currently at 1, I do not see anywhere safe to cut as the shape function simply do not vanish to zero.



The same happens with constant base (B=1)

However i now have a constant of 1/n (where n is the number of points) everywhere not affected 



If I use only RBF then i loose constant and linear reproductivity.

However i seem to have a compact domain.



But i still have a problem which apply to all case:

I'll call it chain of points.

A point is affected by all points in it's distance of influence



However if point B is in the dist of influence of A

And C influence B

And D influence C

And ...





In the system matrix this can be seen as there's no "block" of points for which i can solve the system independently of other block.



On a regular grid this is a big problem

If the distance of influence is big enougth to have another point

Then that point will have at leat another point in it's distance of influence

And the chain can easily cover the whole domain of the problem.





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Did I made an error somewhere ?



From what i see Compactly supported RBF have a band structure in it's system of equation

But it should not lead to a band structure rigidity matrix by itself



(However arbitraily cutting the shape function out of a domain seam to give sort of good result)



Thank you for your answer

- JC