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Trying to perform mesh-to-mesh 3D interpolation on unstructured grid. Mesh size is about 200k nodes. Painful thing that interpolation shoud be repeated ~1000 times (different thermal map) on the same mesh.

One "operation" performs in less than a minute (w/ InterpolationOrder->1) --> but 1000 steps takes 12 hours. I am wondering are there any options to optimize the procedure for unstructured grid and make the whole calculation faster, cause mesh doesn`t change but it takes time to generate Denaunay triangulation at each step.

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  • $\begingroup$ Look up NDSolve`FEM`ElementMeshInterpolation. $\endgroup$ – Michael E2 Sep 25 '15 at 9:17
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    $\begingroup$ If that's no help, then perhaps the question is not clear enough. How is your mesh generated? Can you provide a minimal example? What function are you trying to optimize? Might the optimal values be at the nodes (since you using linear interpolation)? Or did you mean performance-tuning instead of mathematical-optimization? $\endgroup$ – Michael E2 Sep 25 '15 at 9:23
  • $\begingroup$ There are two list of nodes (for example): inputlist=RandomReal[{-10, 10}, {200000, 3}] and outputlist=RandomReal[{-10, 10}, {200000, 3}] (in these example boundaries may be different, but in my case they are equal). Both lists represent unstructured grid. Another list contains data in the inputlist at different timesteps (example): Temp=RandomInteger[{100,200},{200000,1000}]. I made a loop with TempFunc = Interpolation[DataTable, InterpolationOrder -> 1] function, where DataTable=MapThread[List, {inputlist, Temp[[istep]]}]. $\endgroup$ – Artemiy Sep 28 '15 at 8:18
  • $\begingroup$ Related: mathematica.stackexchange.com/questions/68973/… (but the grid has a semi-regular structure). $\endgroup$ – Michael E2 Sep 28 '15 at 10:19
  • $\begingroup$ Thanks, Michael E2. NDSolveFEMElementMeshInterpolation helps. Becomes x4 faster. $\endgroup$ – Artemiy Sep 29 '15 at 15:15
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One can use generate an ElementMesh once and for all (15-16 sec.); then use ElementMeshInterpolation on each coordinate to construct interpolations (1.7 sec.).

Needs["NDSolve`FEM`"]

SeedRandom[0];
inputlist = RandomReal[{-10, 10}, {200000, 3}];
outputlist = RandomReal[{-10, 10}, {200000, 3}];

( mesh = DelaunayMesh[inputlist];  
  elem = Thread[MeshCells[mesh, 3], Tetrahedron] /. 
    Tetrahedron -> TetrahedronElement;
  (*coords = MeshCoordinates[mesh];
  coords == inputlist*)
  emesh = ToElementMesh[
    "Coordinates" -> inputlist,
    "MeshElements" -> {elem}];
 ) // AbsoluteTiming

ifns = ElementMeshInterpolation[{emesh}, #, 
      "ExtrapolationHandler" -> {Indeterminate &, 
        "WarningMessage" -> False}] & /@ 
    Transpose[outputlist]; // AbsoluteTiming
(*
  {15.7423, Null}
  {1.69745, Null}
*)

One can construct various interfaces to the interpolating functions. The matrix one is faster on lists of points, though only by a little.

Clear[ifn];
ifn[v_?VectorQ] := Through[ifns @@ v];
ifn[m_?MatrixQ] := Through[ifns @@ Transpose[m]];
ifn[x_?NumericQ, y_?NumericQ, z_?NumericQ] := Through[ifns[x, y, z]];

Evaluating the interpolating function is not fast, but I assume that is probably true no matter what. (An unstructured grid is handle by an ElementMesh under the hood, AFAIK.) In any case, constructing emesh only once saves a lot of time. I also am getting a small error with my last iteration. (There wasn't any the first time I tried.)

ifn@inputlist[[;; 10]] - outputlist[[;; 10]] // Abs // Max // AbsoluteTiming
(*  {0.584745, 1.06581*10^-14}  *)

But there doesn't seem to be a problem with the data in the interpolating functions:

Table[
 {ifns[[i]]["ElementMesh"]["Coordinates"] == inputlist,
  ifns[[i]]["ValuesOnGrid"] == outputlist[[All, i]]}, {i, 3}]
(*  {{True, True}, {True, True}, {True, True}}  *)

I suppose it's a weakness in the internal interpolating code.

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