I have a problem that can be viewed in two different ways:
Compute an $n$-dimensional integral, numerical context. The domain of integration is an $n$-dimensional hyper-cube of side $L$.
Count (just count) the roots of an $n$-dimensional function (not a polynomial).
Solving just one of them is sufficient for solving the original problem. I know that simple algorithms for numerical integration would take $O(L^n)$, taking linear time per dimension. But I am not sure if there an asymptotically faster algorithms for (1).
For (2), I am aware of algorithms that can find roots (Newton and Bisection), but I am not sure about the best algorithms just for counting how many roots are in a non-polynomial $n$-dimensional function.
What are the best algorithms for (2)? Are they better than the fastest of (1)?