Non-equilibrium allele frequency spectra via spectral methods

A major challenge in the analysis of population genomics data consists of isolating signatures of natural selection from background noise caused by random drift and gene flow. Analyses of massive amounts of data from many related populations require high-performance algorithms to determine the likelihood of different demographic scenarios that could have shaped the observed neutral single nucleotide polymorphism (SNP) allele frequency spectrum. In many areas of applied mathematics, Fourier Transforms and Spectral Methods are firmly established tools to analyze spectra of signals and model their dynamics as solutions of certain Partial Differential Equations (PDEs). When spectral methods are applicable, they have excellent error properties and are the fastest possible in high dimension; see Press et al. (2007). In this paper we present an explicit numerical solution, using spectral methods, to the forward Kolmogorov equations for a Wright-Fisher process with migration of K populations, influx of mutations, and multiple population splitting events.