Is it possible to optimize (watch the code below) NIntegrate?

Questions
,
1

Hi everyone. I’m new in Mathematica and that process take 7-8 hours long. I’ve tried to use Method->“MonteCarlo” and “QuasiMonteCarlo” but it gave nothing at all. Please, give some advice to how can Wolfram count up this process quickly.

intt[\[Phi]_?NumericQ, y_?NumericQ] :=
2*NIntegrate[(
   79*1/137*2/(4/3*3.14*R)*Sinh[y]*E^(y/2)/(4 Sinh[4.197/2])*
    x^2 Re[Sqrt[
     R^2 - (b/2 +
        x)^2]] Piecewise[{{(R^2 - (x + b/2)^2) (R^2 - (b/2 -
            x)^2), (R^2 - (x + b/2)^2) (R^2 - (b/2 - x)^2) >
        0}, {0, (R^2 - (x + b/2)^2) (R^2 - (b/2 - x)^2) <=
        0}}])/(x^2 + ((t + 2)/100 Sinh[y])^2)^(
   3/2), {x, 0,
    Ri/2 (b/Ri Cos[\[Phi]] + Sqrt[4 - (b/Ri Sin[\[Phi]])^2])},
   AccuracyGoal -> 2]

Table[NIntegrate[
  intt[\[Phi], y], {\[Phi], -3.14/2, 3.14/2}, {y, -4.197, 4.197},
  AccuracyGoal -> 2], {t, 298}]
2

It appears you can pull 79*1/137*2/(4/3*3.14*R)*Sinh*E^(y/2)/(4 Sinh[4.197/2]) outside of your NIntegrate and thus potentially avoid having that recalculated many times inside NIntegrate.

It appears you can rewrite Piecewise[{{expr,expression>0},{0,expression<=0}}] as Piecewise[{{expr,expression>0},0}] and save some calculations.

What I think holds most promise is to study the values of R, b, Ri, phi and x to eliminate the Piecewise completely and only integrate over the nonzero part of the integrand. When I look at Reduce[(R^2 - (x + b/2)^2) (R^2 - (b/2 - x)^2) == 0, x] I see that the boundary depends fairly simply on the values of your parameters and I suspect that knowing what the values of your parameters are will make this even simpler..

3

Take a look here, especially the comments and the links within.

The very first things to try is to turn off symbolic processing using Method → {Automatic, “SymbolicProcessing” → 0} For me this speeds up your integral by an order of magnitude. If that’s not enough, the next step is to Compile[] the integrand. I think you’ll need to rewrite the piecewise as an If[] to make sure compilation gives you optimal performance.

EDIT:

Here’s how to compile to integrand for more efficient evaluation:

cf = Compile[{{x, _Real}, {y, _Real}},
  Module[{e1, e2},
   e2 = R^2 - (b/2 + x)^2;
   e1 = e2 (R^2 - (b/2 - x)^2);
   (79 2 x^2 E^(y/2)
     Sinh[y] If[e2 > 0, Sqrt[e2], 0] If[e1 > 0, e1, 0])/(
   1/3 (137 (4 3.14 R) (4 Sinh[4.197/2])) ((1/100 (t + 2) Sinh[y])^2 +
       x^2)^(3/2))],
  
  CompilationOptions -> {"InlineExternalDefinitions" -> True}]

fun[x_?NumericQ, y_?NumericQ] := cf[x, y] (* avoid errors about non-numeric arguments *)

This gives another 2-3 times speedup on my machine compared to just turning off symbolic processing. If you reduce the accuracy goal, as you did, it can be sped up even more significantly.

4

Thanks a lot! That’s just what I need!