Extract symbolically processed form of NIntegrated function

Question

As I understand NIntegrate does some symbolic preprocessing of inputs before it actually goes ahead and numerically integrates. I'm wondering if there is some way to access this intermediate preconditioned form.

Let me demonstrate:

expr[a_, b_, c_] = -((a xp + b yp - c zp)/Sqrt[(0.5` - xp)^2 + yp^2 + zp^2]);
dom = Triangle[{{0.`, 0.`, 1.`}, {0.25, 0.8, 0.4}, {-0.7, 0.5, 0.4}}];
Timing[NIntegrate[expr[1,2,3], {xp, yp, zp} \[Element] dom]]
Timing[NIntegrate[expr[1,2,3], {xp, yp, zp} \[Element] dom, 
  Method -> {Automatic, "SymbolicProcessing" -> 0}]]
{0.03125, 0.474964}
{0.21875, 0.47448}

As you can see there's a healthy performance gain when allowing NIntegrate to precondition its inputs.

I need to compute thousands+ of similar integrals over different domains and different parameters. It would be nice if I could get access to this preconditioned form so I could inject my parameters directly into that or perhaps compile it, rather than duplicating this simplification step each time.

Maybe, you should ask about how to speed up the computation of thousands of similar integrals, not (just) how to get the symbolic pre-processing forms of NIntegrate. — Anton Antonov, Nov 17 '20 at 01:29
@AntonAntonov That sounds like a great idea. https://mathematica.stackexchange.com/questions/234887/speed-up-the-computation-of-many-similar-numerical-integrals-over-2d-regions-emb — alessandro, Nov 18 '20 at 01:22
@AntonAntonov is there a way to adapt your answer here to solving problems of this kind (multi-dimensional)? https://mathematica.stackexchange.com/questions/126001/how-to-calculate-the-numerical-integral-more-efficiently/126041#126041 I noticed that in ArrayOfFunctionsRule.m, multidimensional integration is listed as a todo. — alessandro, Nov 25 '20 at 16:36
Yes, I as thinking to adapt that answer to your problem last week(end), but did not have time to do it. — Anton Antonov, Nov 25 '20 at 16:50

score 6 · Accepted Answer · answered Nov 17 '20 at 01:52

You can use IntegrationMonitor to examine the integrals and regions NIntegrate computes with after the symbolic preprocessing.

Here is an example:

NIntegrate[expr[1, 2, 3], {xp, yp, zp} \[Element] dom, 
 MaxRecursion -> 0, PrecisionGoal -> 1, IntegrationMonitor -> Print]
(* During evaluation of In[35]:= {NIntegrateIntegrationRegion[{{0,0},{1,1}},ExperimentalNumericalFunction[{xp,yp},-((3.21926 (1-xp) (-0.821918+1. xp+0.575342 (1-xp) yp))/Sqrt[1.17647 -1.36471 xp+1. xp^2-0.470588 (1-xp) yp+1.10118 (1-xp) xp yp+1.03529 (1-xp)^2 yp^2]),-NumericalFunctionData-],{},NIntegrate`MultiDimensionalRule[{{{{0.,0.},{0.179284,0.},{0.474342,0.},{0.474342,0.474342},{0.344124,0.344124}},{-0.193873,0.149367,0.0518214,0.0101611,0.0871183},{1.13809,-0.354748,0.00723975,-0.0241325,0.0871183}},{{},{},{},{}},{2,5,-1}}]]} *)
(* 0.474984 *)

Extract symbolically processed form of NIntegrated function

1 Answers1