BibTex format
@unpublished{Gutleb:2020,
author = {Gutleb, TS and Carrillo, JA and Olver, S},
publisher = {arXiv},
title = {Computing equilibrium measures with power law Kernels},
url = {http://arxiv.org/abs/2011.00045v1},
year = {2020}
}
Publications
Citation
RIS format (EndNote, RefMan)
TY - UNPB
AB - We introduce a method to numerically compute equilibrium measures forproblems with attractive-repulsive power law kernels of the form $K(x-y) =\frac{|x-y|^\alpha}{\alpha}-\frac{|x-y|^\beta}{\beta}$ using recursivelygenerated banded and approximately banded operators acting on expansions inultraspherical polynomial bases. The proposed method reduces what is naively adifficult to approach optimization problem over a measure space to astraightforward optimization problem over one or two variables fixing thesupport of the equilibrium measure. The structure and rapid convergenceproperties of the obtained operators results in high computational efficiencyin the individual optimization steps. We discuss stability and convergence ofthe method under a Tikhonov regularization and use an implementation toshowcase comparisons with analytically known solutions as well as discreteparticle simulations. Finally, we numerically explore open questions withrespect to existence and uniqueness of equilibrium measures as well as gapforming behaviour in parameter ranges of interest for power law kernels, wherethe support of the equilibrium measure splits into two intervals.
AU - Gutleb,TS
AU - Carrillo,JA
AU - Olver,S
PB - arXiv
PY - 2020///
TI - Computing equilibrium measures with power law Kernels
UR - http://arxiv.org/abs/2011.00045v1
UR - http://hdl.handle.net/10044/1/88648
ER -