I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that space.

$$\frac{1}{m} \sum f(x_i) \approx \int_{[0,1]^n} f(x) \ dx $$

If I broke each segment $[0,1]$ into $10$ points, I would have to calculate $10^n$ values of my function $f$ - way too many.

Are there lattices I can use which become dense in $[0,1]^n$ as the mesh gets smaller, and whose points do not grow too quickly?

If I knew more about lattices, I could make this more precise.

stratified samplingand you might have some luck searching under that term. $\endgroup$ – Steven Stadnicki Dec 31 '13 at 19:05