Gradient-based Monte Carlo methods for relaxation approximations of hyperbolic conservation laws

TL;DR

The paper tackles the high-variance challenge of Monte Carlo solvers for hyperbolic conservation laws and introduces Gradient-based Monte Carlo (GBMC), built on relaxation approximations and gradient sampling, to create grid-free, variance-reducing particle methods. It demonstrates substantial accuracy gains and sharper shock resolution in 1D scalar problems (e.g., Burgers, LWR) and in hyperbolic systems (SWE, Aw–Rascle, Isentropic Euler) compared with standard MC, including handling negative solutions. The work provides a path toward adaptive probabilistic solvers for hyperbolic PDEs and potential extensions to multidimensional problems and kinetic equations.

Abstract

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

Figures (12)

• Figure 1: Standard normal distribution using $N=100$ samples. Left: histogram of Monte Carlo samples with $M=50$ cells. Middle: histogram of its derivative using two symmetric families of samples with positive and negative masses and $M=50$ cells. Right: plot of the cumulative distribution of the derivative samples using left reconstruction as in \ref{['eq.recL']}.
• Figure 2: Left: The state variable $u$ uniquely defines the number of particles in the two equilibrium states $E^{\pm}$ when the latter have the same sign. Right: There exist infinite possible ways to associate particles to the equilibrium states $E^{\pm}$ when equilibria have discordant sign.
• Figure 3: Convergence rate analysis. Comparison of the relative $L^2$ error norms of the direct Monte Carlo (MC) and the Gradient-based Monte Carlo (GBMC) with respect to the number of particles $N$ for the solution of the inviscid Burgers equation at $t=2.5$ with normal distribution as initial datum (left) and at $t=0.5$ with sinusoidal distribution as initial datum (right).
• Figure 4: Test 1(a), Burgers equation. Solution at $t=10$ with a Gaussian density as initial datum, shown in gray dotted line. Here $a=0.4$ and $\Delta t = 0.5$ (first and third row, left), $\Delta t = 0.25$ (first and third row, right), or $\Delta t = 0.1$ (second and fourth row). First two rows: Standard MC with $N=1000$ particles and $M=50$ grid points (left), and $N=10000$ and $M=100$ grid points (right). Last two rows: GBMC method with $N=100$ (left) and $N=1000$ (right). The subplots refer to the histogram of the space derivative of the state variable. The reference solution is reported in black solid line.
• Figure 5: Test 1(b), Burgers equation. Solution at $t=10$ with a square wave as initial datum, shown in gray dotted line. Here $a=0.6$ and $\Delta t = 0.01$. First row: Standard MC with $N=1000$ particles and $M=50$ grid points (left), and $N=10000$ and $M=100$ grid points (right). Last row: GBMC method with $N=100$ (left) and $N=1000$ (right). The subplots refer to the histogram of the space derivative of the state variable. The reference solution is reported in black solid line.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Lemma 1
• Theorem 1
• Theorem 2
• Lemma 2
• Lemma 3: Glivenko-Cantelli
• Theorem 3