When is Monte Carlo Integration useful?

Question

Monte Carlo Integration appears to be a technique for 'brute forcing' an integral that cannot be otherwise (analytically) solved (or easily solved) without the use of any calculus, with the main drawback I am finding being the large cost in terms of computing power to use the method due to the number of random numbers we need to introduce to find a sufficiently accurate result (hence why I believe it isn't useful for $1$D integrals when we have tools such as Simpson's rule or the Midpoint rule too compute integrals). In what kind of situation would one be content using a Monte Carlo Integration technique despite the cost?

Answer 1

You are right. For most of the 1D integration, the numerical integration performs better than the MC method. For integration, MC method is usually used for high dimensional cases, like 3D integration or even higher than 3D. Even known this, the general rule for integration is like this: if there is analytical solution then stick analytical solution; if numerical solution is doable then use numerical method; if those first two methods fails then consider MC method.

Answer 2

Consider integration over the unit cube in dimension $n $. At a given error level, typical numerical methods need something like $C^n $ points, where $C $ depends only on the error threshold and the (quantitative) regularity of the integrand (e.g. derivative bounds or Fourier bounds). Monte Carlo methods require a constant number of points, depending only on the error and the regularity of the integrand but not the dimension. So they win out in high dimensions. In 1D they are usually a bad choice unless you can use very effective importance sampling.

The situation is a bit more complicated for irregular geometries.