I'm trying to sample a lot of points efficiently. I'm wondering if the following method is possible.
I sample points of a function (evaluate the function) mod $n$. I.e. I calculate f(element one), f(element 2)...f(element n). I do this for all of the points in a ring. Now I take a second ring (mod $m$, for instance) and calculate the points of a function for all elements of this ring.
So my idea is this: I've sampled only $n+m$ points, but the combinations of these sample points number $n \cdot m$. I'd like to calculate the sum of (evaluations of) this larger set of points efficiently. I'm wondering if this is possible.
I'd like to know as much about this idea as possible. I'd like to know related ideas, what's possible, or just general ideas on how to do this. Any pointers would constitute an acceptable answer to me, and an actual method would be absolutely wonderful.
I realize that this may be more of a numerical method than mathematics, but I'm hoping that everyone will accept this question, since it may allow new methods of integration as a result.