Sparse Spectral Methods for Power-Law Interactions

Sheehan Olver
Imperial College

Attractive-repulsive power law equilbriums are an important tool in modelling phenomena in collective behaviour: picture a flock of birds which simultaneously group together, but not too closely (i.e., they practice social distancing), until an equilibrium distribution is reached. In this talk we show that orthogonal polynomials have sparse recurrence relationships for power law (Riesz) kernels. This leads to highly structured and efficiently solvable linear systems for the attractive-repulsive case with two such kernels of opposite sign, giving an effective numerical method for computing such equilibrium distributions. This links to and builds on related work in logarithmic potential theory, singular integral equations, and fractional differential equations.