I'm trying to improve the accuracy of a numerical integral of a very complicated function using the Monte Carlo Method by changing the MaxPoints. While doing that I asked to myself if it is better to perform a lot of (say 10) Monte Carlo integrations and then perform the average of all of them or to just increase the value of MaxPoints by a factor 10.
As an example I have performed 7 Monte Carlo integrations using "MaxPoints = $10^6$"
fourkintegral[fun_, k1_, k2_, k3_, k4_, x3_, x5_, x7_, points_] :=
With[{sol =
NIntegrate[(2 Pi) k1 k2 k3 k4 fun Cos[2 (x3 - x5 - x7)], {k1, 0,
Infinity}, {k2, 0, Infinity}, {k3, 0, Infinity}, {k4, 0,
Infinity}, {x3, 0, 2 Pi}, {x5, 0, 2 Pi}, {x7, 0, 2 Pi},
Method -> {"MonteCarlo", "MaxPoints" -> points},
IntegrationMonitor -> ((errors = Through[#@"Error"]) &) ]},
Around[sol, Total@errors]]
and then I did
fourkintegral[funint[1], k[1], k[3], k[5], k[7], x3, x5, x7, 10^6]
7 times ("funint[1]" is my complicated function) and I performed the average.
The result was
Out[64]= Around[0.0007229541751627402, 0.0004192286316295475]
On the other hand doing the same integral by changing the value of MaxPoints by a factor 7 I obtained the following result
In[66]:= fourkintegral[funint[1], k[1], k[3], k[5], k[7], x3, x5, x7,7*10^6]
Out[66]= Around[0.0003400475222668622, 0.000559829413446022]
The error estimates for the first result is better and therefore I have the following question:
Is it better to perform the average o $N$ Monte Carlo Integrals using $P$ points or just making one integral using $N \times P$ points?
