# Numerical spherical integration

In a high-dimensional setting, say $d \gg 5$, what is a recommended way of evaluating a spherical integral of a smooth (non-symmetric) function $f(\mathbf{x})$?

$\int_\mathcal{S_r} f(\mathbf{x}) \mathrm{d}(\mathbf{x})$

where $\mathcal{S_r} = \{ \mathbf{x} \; | \; \|\mathbf{x}\|^2 = r \}$.

I want to avoid Monte Carlo integration due to high requirement on the number of samples. There seem to be various quadrature methods, but I don't know which one to use. Is there a recent survey of these integrals perhaps?

• why not monte carlo? once $d-1>>1$ it'll be difficult to integrate by other means. Apr 8 '14 at 22:06
• You can look up quasi-Monte Carlo, which has $O(N^{-1})$ convergence instead of $O(N^{-1/2})$, or sparse quadrature. Apr 8 '14 at 23:16
• In high dimensions a standard normal distribution concentrates highly on the surface of the sphere, so if you don't need an exact answer you could integrate $f(x)e^{-|x|^2}dx$ over a box containing the sphere instead of the sphere itself, perhaps using sparse grids. Apr 9 '14 at 2:23
• You can use the SphericalCubature package for R. Mar 3 '18 at 10:14