Low-Complexity Energy Methods for Discretization

Abstract

One potential difficulty that arises in numerical generation of near-optimal configurations is the computational complexity of the energy sums and their derivatives. One way of dealing with this issue is to minimize weighted energy sums, where the weight varies with N and vanishes if the points x and y are further away from each other than a certain threshold distance \(r_N\). A natural question arises whether the limiting distribution of asymptotically optimal sequences of N-point configurations and the leading term of the minimal energy with the varying weight coincide with the ones of the energy with some fixed weight.