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?
Numerical spherical integration
numerical analysis;quadrature