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What are the most sophisticated methods for interpolating a scalar field say Electric or Magnetic Field on a 3-D grid?

I have scalar data on a meshgrid with equal spacing. I would like to use an interpolating method to interpolate those values onto random points I select on the same grid. I think my best bet is using tricubic splines. Am I right? Some of the scalars might not be linear with others being linear though. Any guidance would be appreciated.

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    $\begingroup$ What is best depends on the interpretation of the values you have on your uniform grid. If they were computed with a finite element method, for example, then there is an underlying function space that dictates what the interpolation ought to be. $\endgroup$ – Wolfgang Bangerth Mar 16 at 15:22
  • $\begingroup$ What if that is not known? I understand that knowing the interpolating method used by theprogram i could easily get my data interpolated. What if i dont know it though? $\endgroup$ – Kostis Papadakis Mar 17 at 0:44
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    $\begingroup$ If you really know nothing, then cubic splines seems like a good choice. $\endgroup$ – Wolfgang Bangerth Mar 17 at 23:06

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