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@article{Gutleb:2022:mcom/3740,
author = {Gutleb, T and Carrillo, J and Olver, S},
doi = {mcom/3740},
journal = {Mathematics of Computation},
pages = {2247--2281},
title = {Computing equilibrium measures with power law kernels},
url = {http://dx.doi.org/10.1090/mcom/3740},
volume = {91},
year = {2022}
}

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TY  - JOUR
AB  - We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the formKpx ´ yq “ |x´y|αα ´|x´y|ββusing recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomialbases. The proposed method reduces what is na¨vely a difficult to approachoptimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibriummeasure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimizationsteps. We discuss stability and convergence of the method under a Tikhonovregularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, wenumerically explore open questions with respect to existence and uniquenessof equilibrium measures as well as gap forming behaviour in parameter rangesof interest for power law kernels, where the support of the equilibrium measuresplits into two intervals.
AU  - Gutleb,T
AU  - Carrillo,J
AU  - Olver,S
DO  - mcom/3740
EP  - 2281
PY  - 2022///
SN  - 0025-5718
SP  - 2247
TI  - Computing equilibrium measures with power law kernels
T2  - Mathematics of Computation
UR  - http://dx.doi.org/10.1090/mcom/3740
UR  - https://www.ams.org/journals/mcom/2022-91-337/S0025-5718-2022-03740-6/home.html
UR  - http://hdl.handle.net/10044/1/95127
VL  - 91
ER  -

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