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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?

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    $\begingroup$ why not monte carlo? once $d-1>>1$ it'll be difficult to integrate by other means. $\endgroup$
    – Aksakal almost surely binary
    Apr 8, 2014 at 22:06
  • $\begingroup$ You can look up quasi-Monte Carlo, which has $O(N^{-1})$ convergence instead of $O(N^{-1/2})$, or sparse quadrature. $\endgroup$
    – Daniel Shapero
    Apr 8, 2014 at 23:16
  • $\begingroup$ 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. $\endgroup$
    – Nick Alger
    Apr 9, 2014 at 2:23
  • $\begingroup$ You can use the SphericalCubature package for R. $\endgroup$
    – Stéphane Laurent
    Mar 3, 2018 at 10:14

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I'd use one of the hypersphere random point pickers and integrate with Monte Carlo.

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