# Monte Carlo integration - convergence

I have a 5D integral being calculated with a Monte Carlo uniform random sampling. The issue is that the region of integration is very small and for 100000 points I get only around 20-30 points every time. The function has to be calculated several times for a certain parameter, which doesn't change the integration region, but changes only the magnitude of the result (which is good, I suppose).

After several runs I gathered some hundreds of points which are a valid match for the integration region. I was thinking about use those points to improve the integration result, but how can I do that, in a correct way?

• Are you actually drawing from $\mathcal U(0,\,1)$ and checking if the point is in the region? Or do you find $\mathcal U(a,\,b)$? – Kyle Kanos Sep 18 at 11:27
• what's U(0,1) ? Let's say for example I'm drawing points from $x \in [0,10]$ and $y \in [0,5]$ and the condition is $x^2 + y^2 < 1$. I'm keeping all the points that respect this condition, because they are so few. – LowFieldTheory Sep 18 at 12:21
• $\mathcal U(0,\,1)$ is standard terminology for "uniform random number between 0 & 1." your constraint for $x,\,y$ makes it impossible 90% of the time simply because of the radicand can't be negative ($y=\sqrt{1-x}$ only works if $x\leq1$) – Kyle Kanos Sep 18 at 12:27
• @KyleKanos of course, that was only a representative example, the conditions here are more complicated. I'm trying to show that the points are difficult to find, that's the essence of the question. – LowFieldTheory Sep 18 at 12:31
• So if that's what is going on, you need to find a way to adjust the parameters. Without being more explicit in the parameters, ranges & equations, it is unlikely anyone can help beyond my first sentence in this comment. – Kyle Kanos Sep 18 at 13:05