Maximum likelihood discretization of the transport equation

TL;DR

This paper introduces maximum likelihood discretization (MLD) for the transport equation, replacing traditional Galerkin methods to preserve positivity and providing theoretical error bounds in Kullback-Leibler divergence.

Contribution

It proposes a novel MLD approach that replaces the method of moments with maximum likelihood, leading to positivity-preserving discretizations and a new Fisher-Rao Galerkin method.

Findings

FRG preserves positivity empirically.

Error bounds established in Kullback-Leibler divergence.

MLD improves stability over traditional methods.

Abstract

The transport of positive quantities underlies countless physical processes, including fluid, gas, and plasma dynamics. Discretizing the associated partial differential equations with Galerkin methods can result in spurious nonpositivity of solutions. We observe that these methods amount to performing statistical inference using the method of moments (MoM) and that the loss of positivity arises from MoM's susceptibility to producing estimates inconsistent with the observed data. We overcome this problem by replacing MoM with maximum likelihood estimation, introducing $\textit{maximum likelihood discretization} $(MLD). In the continuous limit, MLD simplifies to the Fisher-Rao Galerkin (FRG) semidiscretization, which replaces the $L^2$ inner product in Galerkin projection with the Fisher-Rao metric of probability distributions. We show empirically that FRG preserves positivity. We prove rigorously that it yields error bounds in the Kullback-Leibler divergence.