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I'm looking for the adequate numerical interpolation technique to solve the following problem. This is probably trivial for physicists who study gravitational fields, but I didn't find clear answers in the literature.

I'm in a high dimensional space (theoretically infinite but numerically finite). I have a fair number of trajectories in this space from which I extract acceleration with second order forward finite differences. Now I want to smoothly interpolate these local accelerations to obtain a "gravitational like" force field in this space.

As the dimensionality is pretty high, I would tend to choose a mesh-free technique, and at first I was going for a vector version of RBF interpolation, or at least a component-wise version. Moreover, as the data is unevenly distributed, that seemed like a good solution. I found some sources explaining that RBF interpolation is not the right method for gravitational force field interpolation, because it is too "local". So I'm still looking for the state of the art technique for this problem.

I assume this type of interpolation is quite common in gravitational physics in low dimension and that the problem generalizes well to high dimension. But even for low dimensional spaces, I can't find any decisive source.

Anything in mind ?

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    $\begingroup$ Hi and welcome to the Math.SE: I leave this comment just to precise that. RBF interpolation means "Radial Basis Function interpolation". Not being a specialist in interpolation theory, I think it adds some information to your nice question $\endgroup$
    – Daniele Tampieri
    Commented Jan 17, 2023 at 6:44
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    $\begingroup$ Your question doesn't actually say what you mean by "gravitational interpolation". $\endgroup$
    – Igor Khavkine
    Commented Jan 17, 2023 at 6:53
  • $\begingroup$ I've noticed that you edited the phrase "gravitational interpolation" to "gravitational force field interpolation". However, that doesn't really change anything. You need a mathematical formulation of what you want for the question to be well-posed. $\endgroup$
    – Igor Khavkine
    Commented Jan 17, 2023 at 9:23
  • $\begingroup$ Thank you for your comment ! Actually I just need the name of a known general method (or the confirmation it doesn't exist) for the problem that can be somehow summarized like: Given local gravitational field, find the most accurate smooth interpolation in the whole space in the sense of classical physics. $\endgroup$
    – Youcef
    Commented Jan 17, 2023 at 10:04
  • $\begingroup$ RBF interpolation is "local" if you choose compactly supported basis functions, but others are available (though do result in dense matrix systems to solve, so limit the number of sources you can handle, practically). $\endgroup$
    – J.J. Green
    Commented Jan 17, 2023 at 10:13

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