Maximum likelihood discretization of the transport equation

TL;DR

This paper addresses the loss of positivity in Galerkin discretizations of transport equations by reframing the time-evolution step as a maximum likelihood estimation problem, yielding a maximum likelihood discretization (MLD). In the small-time-step limit, MLD reduces to the Fisher-Rao Galerkin (FRG) semidiscretization, which replaces the $L^2$ inner product with a Fisher-Rao–weighted inner product, and to its discontinuous DG variant (DFRG) that enforces local conservation. The authors prove KL divergence-based error bounds for FRG/DFRG and show, through extensive 1D and 2D numerical experiments, that FRG preserves positivity and achieves competitive or superior KL-based accuracy compared to traditional Galerkin methods with positivity limiters. The work positions FRG/DFRG as robust, information-geometric alternatives for transport discretization and outlines future extensions to other conservation laws and divergences. Overall, the approach combines statistical inference, information geometry, and conservative discretization to yield positivity-preserving, KL-controlled transport solvers with practical effectiveness.

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.

Figures (16)

• Figure 1: The solution leaves the Galerkin space (shown as mesh), even if the initial condition is contained in it.
• Figure 1: Density profile for \ref{['ex:1D-mild-compression']}, a 1D sinusoidal problem with polynomial order $p=1$, constant initial density, and fixed velocity $u(x) = sin(2 \pi x) + 2.0$. The solution is obtained using the DG and DFRG methods with CFL = 0.1875 and $m=256$ cells. The profiles are shown at (a) $t=0$, (b) $t=2$, and (c) $t=3$.
• Figure 2: Galerkin projection of the solution to transport equation after each time step $\Delta t$.
• Figure 2: Error convergence under grid refinement for \ref{['ex:1D-mild-compression']}, a 1D sinusoidal problem with polynomial order $p=1$, constant initial density, and fixed velocity $u(x) = sin(2 \pi x) + 2.0$. The mean of the (a) $L^{1}$, (b) $L^{2}$, and (c) $KL$ errors with respect to the exact solution is measured at points spaced out by $\Delta t=0.01$ up to time $T=3$, for the DG, DG $+$-limiter and DFRG methods.
• Figure 3: A 2D simplex representing subset of feasible probability distributions, intersected by 2D subset of Galerkin space, which contains feasible approximate solutions. A solution initial in the Galerkin space exits it after a time-step, requiring projection back onto the Galerkin space. In this example, the $L^{2}$ projection may leave the simplex (violating probability constraints), while a Fisher-Rao (FR) projection remains in the simplex structure, ensuring a valid probability distribution.

Theorems & Definitions (12)

• Remark 2.1
• Remark 4.1
• Remark 4.2
• Definition 5.1
• Lemma 5.2
• Proof 1
• Corollary 5.3
• Proof 2
• Theorem 5.4
• Proof 3