Why is numerical integration hard / infeasible in high dimensions?
TL;DR: the curse of dimensionality.
- If you keep a fixed resolution (say 10 points per axis), grid/quadrature needs 10^d function evals. Even d=50 ⇒ 10^{50} — impossible.
- High-dim geometry is weird: most volume sits near the corners/shells, so “important” regions can be an exponentially tiny fraction of the domain; naive samples almost never land there.
- Space becomes sparse: distances concentrate, so covering the domain well requires exponentially many points.
- Monte Carlo avoids grids but only converges as O(N^{-1/2}); when the integrand’s mass is in tiny regions, variance blows up and you still need (effectively) exponentially many samples without smart importance sampling.