Optimal Monte Carlo Integration on Closed Manifolds

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.