Fast high-dimensional node generation with variable density

Abstract

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

Asymptotics of $k$-nearest neighbor Riesz energies

“…Of course, the knowledge of the C k s,d constant is required, but its value can easily be approximated numerically and is stable with respect to the computation error. This justifies the application of gradient flow to nearest neighbor truncation of the Riesz energy as a means to obtain a prescribed distribution, a strategy previously used as a heuristic [25].…”

“…One approach [3] to reducing this cost involves radial truncation. Instead, here we analyze truncation of E s to a fixed number k of nearest neighbors, as used heuristically in [25]. An advantage of this technique, in contrast to the radial truncation, is that memory and computational costs depend only on k and N (essentially kN ) and not on ω N .…”

“…It has been demonstrated that the energies E k s can be used for efficient discretization of complicated surfaces, see [25,24]. Here we illustrate the effectiveness of the algorithm in Figure 2 which shows an approximate minimizing configuration of N = 20 000 points on an algebraic surface with s = 4, k = 30, V ≡ 0, and with a nonuniform weight.…”

Fast variable density 3-D node generation

“…Recent work has been done on producing quality node sets specifically for RBF-FD [3,4,5,6]. Here we extend the method of 2-D node generation from Fornberg & Flyer [4] to higher dimensions.…”

“…This is, however, computationally wasteful since RBF-FD methods make no use of the often costly step of connecting nodes into good aspect ratio elements. Iterative methods begin with an initial node set and update their positions through either a form of energy minimization [7], short-range interaction forces [8] or gradient flow [6]. These methods are strongly dependent on their initial configuration and can be costly.…”