Distributing points \(\{x_i\}_{i=1}^n\) on the sphere as to minimize the mean square error
\[\mathbb{E}\left[\left(q_n(f) - \int_{\mathbb{S}^2}f(s)\,ds\right)^2\right]\]
of the quadrature formula \(q_n(f) =\frac{1}{n}\sum_{i=1}^n f(x_i)\), where \(f\) is a centered Gaussian process with covariance function \(C(x,y) = \exp(\langle x, y \rangle)\). Shown is \(n=6, 12, 23\).
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Olivier Binette