Why does Monte Carlo integration underestimate my integral as integration range increases?

Question

I'm having a huge problem with Monte Carlo integration. This simple example adequately shows the problem.

ParallelTable[
  NIntegrate[
    3 Sin[θ2] Sqrt[(E^w2/((1 - E^w2 Cos[θ2])^2 + E^(2 w2) Sin[θ2]^2))]
      Sin[θ3] Sqrt[(E^w3/((1 - E^w3 Cos[θ3])^2 + E^(2 w3) Sin[θ3]^2))], 
    {w2, -W, W}, {θ2, 0, Pi}, {w3, -W, W}, {θ3, 0, Pi}, 
    Method -> "QuasiMonteCarlo"] /
  (NIntegrate[
     Sin[θ2]Sqrt[(E^w2/((1 - E^w2 Cos[θ2])^2 + E^(2 w2) Sin[θ2]^2))], 
     {w2, -W, W}, {θ2, 0, Pi},
     Method -> "QuasiMonteCarlo"])^2, 
  {W, 1, 10000, 100}]

This gives

{3.0014, 2.9491, 2.85273, 2.94142, 3.13837, 3.3748, 3.59702, 3.76732, 3.866, 
 3.88818, 3.83872, 3.72786, 3.56807, 3.37201, 3.15141, 2.91649, 2.67575, 2.43598, 
 2.20243, 1.97891, 1.76811, 1.5717, 1.39061, 1.22513, 1.07511, 0.940038, 0.819179, 
 0.711636, 0.616418, 0.532496, 0.458835, 0.394427, 0.338305, 0.28956, 0.247351, 
 0.2109, 0.179505, 0.152529, 0.129403, 0.109619, 0.0927273, 0.0783328, 0.0660878, 
 0.0556886, 0.046871, 0.0394056, 0.033094, 0.027765, 0.0232714, 0.0194869, 
 0.0163033, 0.0136281, 0.0113825, 0.00949942, 0.00792187, 0.00660148, 0.00549731, 
 0.00457473, 0.00380451, 0.00316199, 0.00262639, 0.00218025, 0.00180887, 0.00149994, 
 0.00124312, 0.00102975, 0.000852585, 0.000705567, 0.000583633, 0.000482557, 
 0.000398814, 0.000329467, 0.000272068, 0.000224581, 0.000185313, 0.000152854, 
 0.000126035, 0.000103886, 0.0000856005, 0.0000705103, 0.0000580618, 0.0000477963, 
 0.0000393339, 0.0000323603, 0.0000266154, .0000218844, 0.0000179894, 0.0000147838, 
 0.0000121462, 9.97676*10^-6, 8.19278*10^-6, 6.7262*10^-6, 5.52086*10^-6, 
 4.53049*10^-6, 3.71696*10^-6, 3.04885*10^-6, 2.5003*10^-6, 2.05002*10^-6, 
 1.6805*10^-6, 1.37731*10^-6}

The integral in the numerator is just the integral inside the denominator squared. So this expression is essentially computing $3\,x\,/\,x$. Monte Carlo integration should be calculating the numerator integral to have the same value as the denominator integral squared regardless of how large W is. Thus, you would expect the output to be a list of numbers near 3 if not exactly 3.

But as the results below show, as W increases the Monte Carlo integration becomes worse and worse at estimating the numerator. It progressively underestimates the value of the numerator as W becomes larger and larger. If anyone can help me successfully integrate these types of functions for large W, I would greatly appreciate it.

What I describe here happens even when I use regular Monte Carlo integration and not just Quasi-Monte Carlo.

Answer 1

If you slightly modify the NIntegrate range specifications of your loop variable W then you'll get results you expect:

AbsoluteTiming[
 Block[{pg = 4}, 
  res = Table[
    NIntegrate[
      3 Sin[\[Theta]2] Sqrt[(E^
           w2/((1 - E^w2 Cos[\[Theta]2])^2 + 
            E^(2 w2) Sin[\[Theta]2]^2))] Sin[\[Theta]3] Sqrt[(E^
           w3/((1 - E^w3 Cos[\[Theta]3])^2 + 
            E^(2 w3) Sin[\[Theta]3]^2))], {w2, -W, -10, 10, 
       W}, {\[Theta]2, 0, Pi}, {w3, -W, -10, 10, W}, {\[Theta]3, 0, 
       Pi}, Method -> "MonteCarlo", 
      PrecisionGoal -> 
       pg]/(NIntegrate[
        Sin[\[Theta]2] Sqrt[(E^
             w2/((1 - E^w2 Cos[\[Theta]2])^2 + 
              E^(2 w2) Sin[\[Theta]2]^2))], {w2, -W, -10, 10, 
         W}, {\[Theta]2, 0, Pi}, Method -> "MonteCarlo", 
        PrecisionGoal -> pg])^2, {W, 1, 10000, 100}]
  ]
 ]

(* {15.1931, {2.19657, 3.05144, 2.99393, 3.03217, 3.01784, 
  3.08816, 3.14745, 2.91172, 3.1147, 3.02755, 2.92361, 2.98091, 
  3.18165, 3.06744, 3.0053, 3.02033, 3.03638, 2.97478, 2.91792, 
  2.89669, 2.99929, 3.09445, 3.05545, 3.03786, 2.97139, 3.07607, 
  3.00059, 3.09282, 2.90901, 3.04438, 3.00764, 3.03369, 2.98443, 
  2.93917, 2.96088, 3.03941, 3.06992, 2.9463, 3.03358, 2.88547, 
  3.04671, 2.89731, 2.91739, 2.99103, 3.05259, 3.15771, 2.87781, 
  3.02416, 3.09235, 2.91561, 2.92959, 3.03615, 3.09252, 2.90106, 
  3.06102, 3.01301, 3.05854, 2.9825, 3.01436, 2.95519, 2.88678, 
  3.15286, 3.01529, 2.92353, 3.04329, 2.98486, 3.05863, 2.94734, 
  3.05641, 2.88573, 2.99265, 3.09635, 3.02641, 2.98033, 2.93819, 
  2.96236, 3.04147, 2.90924, 2.90145, 3.00857, 2.98786, 3.16616, 
  2.97454, 3.05536, 2.99765, 2.92687, 3.01301, 2.94392, 2.95414, 
  3.05081, 2.97163, 2.98609, 2.86618, 2.96695, 3.04867, 3.04276, 
  2.98415, 3.05364, 3.07536, 3.04569}}  *)