Optimal Monte Carlo integration on closed manifolds
For researchers in numerical integration on manifolds, this provides a theoretical justification for re-weighting to improve Monte Carlo convergence rates.
This paper shows that re-weighting random points in Monte Carlo integration achieves optimal approximation rates (up to a logarithmic factor) for Sobolev spaces on closed Riemannian manifolds, with numerical experiments on the unit sphere and Grassmannian manifold confirming the theory.
The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $n^{-1/2}$. However, re-weighting of random points can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the unit sphere and on the Grassmannian manifold. Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex.