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enter image description here Given a 2D irregular spaced data like shown in the figure, I would like to know how to find derivatives at '*' by interpolating the values at 'o'. Does lagrange 2D interpolation work at irregular spaced data

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    $\begingroup$ You can triangulate the points and use a finite element type approximation. However you may need to do some extra work to get an accurate gradient. Search for gradient recovery techniques. Another option is to use radial basis functions, which can give very accurate approximations. This is pretty easy to implement. $\endgroup$
    – cfdlab
    Aug 31, 2017 at 12:36
  • $\begingroup$ Yes, you can use Lagrange interpolation for irregular data. Where do you want to evaluate the derivatives? $\endgroup$
    – nicoguaro
    Aug 31, 2017 at 18:47
  • $\begingroup$ @PraveenChandrashekar thanks, i am using gaussian radial basis functions using point interpolation method. I'm able to interpolate the derivatives at uniform grids with error 1e-3. does it give accurate derivatives at non-uniform grids also. or can you direct me any reference. $\endgroup$
    – Neutron17
    Sep 1, 2017 at 13:08
  • $\begingroup$ @nicoguaro i added the figure. can i interpolate derivatives at '*' using function values at 'o' by lagrange interpolation. Thanks $\endgroup$
    – Neutron17
    Sep 1, 2017 at 13:10
  • $\begingroup$ @Neutron17I have not kept up with literature on RBF so cant give good answer. But I have seen people use RBF for solving PDE. I think they just differentiate the interpolant to get derivative approximations. See e.g., papers of Natasha Flyer. $\endgroup$
    – cfdlab
    Sep 1, 2017 at 14:22

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It seems that your mesh is a Voronoi diagram. In that case, "Natural Neighbors Interpolation" [1] may be a "natural" choice (depending on how you construct the mesh and what you want to do with the derivatives). See also the Wikipedia entry on Natural Interpolation [2]. The interpolant is smooth (it is $C^1$ everywhere except on the data points where it is $C^0$, see [3] and references herein). Derivatives can be evaluated, but computations are a bit involved. However, if the points were you want to evaluate the derivatives are the centers of the Delaunay edges (as it seems to be in your figure), then there are good chances that the expression of the derivatives can be simplified.

[1] Sibson, R. (1981). "A brief description of natural neighbor interpolation (Chapter 2)". In V. Barnett. Interpreting Multivariate Data. Chichester: John Wiley. pp. 21–36

[2] https://en.wikipedia.org/wiki/Natural_neighbor_interpolation

[3] http://www-umlauf.informatik.uni-kl.de/~bobach/work/publications/dagstuhl06.pdf

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