Computing Equilibrium Measures with Power Law Kernels

Abstract

We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form Kpx ´yq " |x´y| α α ´|x´y| β β using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases. The proposed method reduces what is naïvely a difficult to approach optimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibrium measure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimization steps. We discuss stability and convergence of the method under a Tikhonov regularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, we numerically explore open questions with respect to existence and uniqueness of equilibrium measures as well as gap forming behaviour in parameter ranges of interest for power law kernels, where the support of the equilibrium measure splits into two intervals.

Computation of Power Law Equilibrium Measures on Balls of Arbitrary Dimension

“…While this is known from numerical particle swarm simulations, there are as of now very few analytic results on this gap formation phenomenon and to our knowledge the only numerical results dealing with the continuous gap formation is the one-dimensional ultraspherical spectral method described in [27]. The present paper sets the foundations for studying these phenomena in higher dimensions and can be seen as an arbitrary dimensional generalization of [27], with many of the obtained results mirroring the one-dimensional case. As such, the method is also related to [42], in which the last author introduced a Chebyshev spectral method for the computation of potential theory equilibrium measures with kernels K 0 " logp|x´y|q.…”

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We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in [27] for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark contrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension.

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“…In the simplest cases, the support of ρ is just a single interval in one-dimension [15,27], a disk in two dimensions and a d-dimensional ball in higher dimensions [15]. For different, in particular higher, powers α and β different support domains for the equilibrium measure may be observed when performing particle simulations, such as two interval systems in one-dimension [15,27], annuli in two dimensions and hyperspherical shells in higher dimensions [15].…”

“…In the simplest cases, the support of ρ is just a single interval in one-dimension [15,27], a disk in two dimensions and a d-dimensional ball in higher dimensions [15]. For different, in particular higher, powers α and β different support domains for the equilibrium measure may be observed when performing particle simulations, such as two interval systems in one-dimension [15,27], annuli in two dimensions and hyperspherical shells in higher dimensions [15]. While this is known from numerical particle swarm simulations, there are as of now very few analytic results on this gap formation phenomenon and to our knowledge the only numerical results dealing with the continuous gap formation is the one-dimensional ultraspherical spectral method described in [27].…”

“…For different, in particular higher, powers α and β different support domains for the equilibrium measure may be observed when performing particle simulations, such as two interval systems in one-dimension [15,27], annuli in two dimensions and hyperspherical shells in higher dimensions [15]. While this is known from numerical particle swarm simulations, there are as of now very few analytic results on this gap formation phenomenon and to our knowledge the only numerical results dealing with the continuous gap formation is the one-dimensional ultraspherical spectral method described in [27]. The present paper sets the foundations for studying these phenomena in higher dimensions and can be seen as an arbitrary dimensional generalization of [27], with many of the obtained results mirroring the one-dimensional case.…”