My MMSC dissertation on general kernel spectral methods for equilibrium measures.
👉 The document itself can be found here!
Attractive-repulsive equilibrium measure problems appear in the modeling of the continuous limit of pair-interacting particle systems, e.g. models of animal swarms or classical physical particulates. From a computing point of view they are a combination of integral equation and minimization problem – the aim is to find a density u(x) for a given kernel K such that the following energy is minimized.
The project would leverage kernel expansions to construct a general equilibrium measure method on the unit interval [−1, 1]. The d-dimensional unit ball generalization is then expected to be straightforward and can be considered as a stretch goal. Comparisons should be made with the corresponding attractive-repulsive finite particle swarm problems.