We present an algorithm for producing discrete distributions with a prescribed nearestneighbor distance function. Our approach is a combination of quasi-Monte Carlo (Q-MC) methods and weighted Riesz energy minimization: the initial distribution is a stratified Q-MC sequence with some modifications; a suitable energy functional on the configuration space is then minimized to ensure local regularity. The resulting node sets are good candidates for building meshless solvers and interpolants, as well as for other purposes where a point cloud with a controlled separation-covering ratio is required. Applications of a three-dimensional implementation of the algorithm, in particular to atmospheric modeling, are also given.where φ(·) is a radial function, and x k , k = 1, . . . , K, is a collection of pairwise distinct points in R d . A common choice of φ is the Gaussian φ(r) = e −( r) 2 , although one may also use 1/(1 + ( r) 2 ), r 2p log(r), p ∈ N, etc. In this discussion, we are not concerned with the distinctions between the
Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as N → ∞) for the Riesz energy of N -point configurations on the d-dimensional unit sphere in the so-called hypersingular case; i.e, for non-integrable Riesz kernels of the form |x − y| −s with s > d. As a consequence, we immediately get (thanks to the Poppy-seed bagel theorem) lower estimates for the large N limits of minimal hypersingular Riesz energy on compact d-rectifiable sets. Furthermore, for the Gaussian potential exp(−α|x − y| 2 ) on R p , we obtain lower bounds for the energy of infinite configurations having a prescribed density.arXiv:1804.05237v1 [math-ph]
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