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The following threads call our attention to the fact that "accuracy of Interpolation will be worse with an unstructured grid", in the context of interpolation of 3d data:

interpolation of 3D data

Improved interpolation of mostly-structured 3d data

I want to know if there is a systematic way to do the interpolation correctly for a 3D data with the points distributed logarithmically along x and y.

Simple example

The xy grid would be built using

grid = Table[ {10^t, 10^u}, {t, 1, 3, 0.05}, {u, 1, 3, 0.1}]
ListPlot[ grid ]

Logarithmic grid

Of course, in logarithmic scale it looks fine (linear):

ListLogLogPlot[ grid ]

enter image description here

The 3D data would be built using some 2 variable function func:

tabx = Table[10^w, {w, 1, 3, 0.05}]
taby = Table[10^w, {w, 1, 3, 0.1}]
tab = Table[ {{x,y},func[x, y]}, {x, tabx}, {y, taby}]

Then, how to interpolate tab properly?

share|improve this question
    
Just using Interpolation works fine and gives high-order interpolation, although you have to use Flatten[tab, 1] to get rid of the nesting. It seems Mathematica still considers your data a structured grid, because it still has the structure of a full grid even though the $x$ and $y$ coordinates are nonuniformly sampled. –  Rahul Narain Feb 15 at 19:45
    
link This was very helpful to me. And I guess just find the log to get the uniform grid then organize the data as shown in the link. –  Lina Feb 16 at 1:10
    
I don't see any difference between using Interpolation and ListInterpolaton for multidimensional data. Is there any difference? Maybe I am missing something. –  fcpenha Feb 16 at 2:02
    
Nevertheless, your point about doing the interpolation on the uniform grid is good. –  fcpenha Feb 16 at 2:49
    
Sometimes I wonder why I bother to leave comments. –  Rahul Narain Feb 16 at 8:27
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