Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation

Abstract

It is known that discrete sets of uniformly distributed points on the hypersphere $\mathbb {S}^{d}\subset \mathbb {R}^{d+1}$ can be obtained from minimizing the energy functional corresponding to Riesz s-kernels $k_s(\boldsymbol {x},\boldsymbol {y})=\lVert \boldsymbol {x}-\boldsymbol {y}\rVert ^{-s}$ (s > 0) or the logarithmic kernel $k_{\log }(\boldsymbol {x},\boldsymbol {y})=-\log \lVert \boldsymbol {x}-\boldsymbol {y}\rVert +\log 2$. We prove the same for the kernel $k_{\log }(\boldsymbol {x},\boldsymbol {y})=\lVert \boldsymbol {x}-\boldsymbol {y}\rVert (\log {\frac {\lVert \boldsymbol {x}-\boldsymbol {y}\rVert }{2}}-1)+2$ which is a front-extension of the sequence of derivatives $k_{\log }, k_{1}, k_{2}, k_{3}, \dots$, up to sign and constants. The boundedness of the kernel simplifies the classical potential-theoretical proof of the asymptotic uniformity of the point distributions. Still, the property of a singular derivative for x → y is preserved, with the physical interpretation of infinite repulsive forces for touching particles. The quality of the resulting point distributions is exemplary compared with that of Riesz- and classical logarithmic point sets, and found to be competitive. Originally motivated by problems of high-dimensional data, the applicability of $\log$-optimal point sets with a novel concentric interpolation and differentiation scheme is demonstrated. The method is significantly optimized by the introduction of symmetrized kernels for both the generation of the minimum energy points and the spherical basis functions. Both the point generation and the Concentric Interpolation software are available as Open Source software and selected point sets are provided.

Appendices

Appendix 1. Generation of random points on $\mathbb {S}^d$

The generation of uniformly random points on $\mathbb {S}^d$, can be realized in several ways. Two prominent and convenient examples include:

• Generation using the normal distribution${\mathcal N}$ (cf. [23]) The easiest way of generating the sought-after points is based on random variables n_i following a normal distribution $\mathcal {N}$. A point $\boldmath {x}\in \mathbb {S}^{d}$ is obtained by setting the components of its coordinate vector $x_i \sim \mathcal {N}$ ($i=1, \dots , d+1$), followed by normalization of the whole vector. A purely technical improvement of the algorithm is to abandon random samples which have a vector norm close to machine precision before the normalization in order to prevent numerical truncation.

• Generation using the uniform distribution${\mathcal U}$on [− 1, 1] Another option is to seed all components of candidate points according to a uniform distribution ${\mathcal U}$ on the interval [− 1, 1]. Next, points having a norm greater than 1 and smaller than a tolerance determined by machine precision are rejected, i.e., only points contained in a spherical shell are accepted. The remaining points are then projected onto $\mathbb {S}^{d}$.

The advantage of the method relying on random variables following a normal distribution is that there is virtually no rejection while the second algorithm will lead to a substantial amount of rejected points: for d = 1 the chance for rejection is 21.46%, for d = 2 it is 47.64% and for arbitrary spherical dimension d it is defined by

$$ \begin{array}{@{}rcl@{}} P(\text{candidate is rejected})=1 - \frac{L^{d+1}(\mathbb{ B}^{d+1} )}{ 2^{d+1} }\text{,} \end{array} $$

(74)

where L^d+ 1 denotes the Lebesgue measure in $\mathbb { R}^{d+1}$ and $\mathbb { B}^{d+1}=\{\boldmath { x}\in \mathbb { R}^{d+1}:\lVert \boldmath { x}\rVert \leq 1\}$. Since Gaussian random variables are available at little numerical expense in many software libraries, the first algorithm is available as an option in the graphical user interface of our MATLAB software. If it is selected, the resulting point set is passed to Algorithm 1 as initial configuration.

Appendix 2. Algorithms

For Algorithm 2, the quantity $\widetilde {f}_{\gamma }$ denotes the surrogate model $\widetilde {f}$ (cf. (CI)), with respect to the kernel function $\widetilde {k}(z_{\mathrm {g}})=\exp (-\gamma z_{\mathrm {g}}^2)$, approximating the original function f.