Multi-fidelity and multi-level Monte Carlo methods for kinetic models of traffic flow

TL;DR

This work addresses uncertainty in kinetic traffic flow models governed by space-homogeneous Boltzmann-type equations and solves the resulting stochastic problem with non-intrusive Monte Carlo methods. It introduces variance-reduction strategies—multi-fidelity and multi-level Monte Carlo—based on low-fidelity surrogates such as approximated steady states and BGK-type models, paired with a Direct Simulation Monte Carlo solver for the kinetic equation. Numerical results show substantial accuracy gains over standard MC, with multi-fidelity methods sometimes outperforming MLMC, especially near the Fokker-Planck limit; performance depends on the chosen surrogate and regime (e.g., small $\varepsilon$). The findings provide a practical framework for efficient uncertainty quantification in kinetic traffic models and suggest directions for extending the approach to non-homogeneous, spatially varying settings and hybrid MF-ML strategies.

Abstract

In traffic flow modeling, incorporating uncertainty is crucial for accurately capturing the complexities of real-world scenarios. In this work we focus on kinetic models of traffic flow, where a key step is to design effective numerical tools for analyzing uncertainties in vehicles interactions. To this end we discuss space-homogeneous Boltzmann-type equations, employing a non intrusive Monte Carlo approach both on the physical space, to solve the kinetic equation, and on the stochastic space, to investigate the uncertainty. To address the high dimensional challenges posed by this coupling, control variate approaches such as multi-fidelity and multi-level Monte Carlo methods are particularly effective. While both methods leverage models of varying accuracy to reduce computational demands, multi-fidelity methods exploit differences in model fidelity, while multi-level methods utilize a hierarchy of discretizations. Numerical simulations indicate that these approaches provide substantial accuracy improvements over standard Monte Carlo methods. Moreover, by using appropriate low-fidelity surrogates based on approximated steady state solutions or simplified BGK interactions, multi-fidelity methods can outperform multilevel Monte Carlo methods.

Figures (7)

• Figure 1: MLMC approach. Expected value of the kinetic density $\mathbb{E}[f]$ for $\varepsilon=0.003$ at different times. Solution computed by Monte Carlo (yellow dashed line), MLMC with $L=2$ (red dashed dotted line), MLMC with $L=3$ (blue line), steady state (green dotted line) and reference solution (black dashed dotted line), at times $T=\frac{1}{4}T_f$ (left) and $T=T_f$ (right), for $z \sim \mathcal{U}([1,3])$.
• Figure 2: MLMC approach. Comparison between the relative errors, computed following \ref{['eq:err']}, of the Monte Carlo method (blue line), MLMC with $L=2$ (black dashed line) and MLMC with $L=3$ (red dotted line) for $\varepsilon=1$ (left) and $\varepsilon=0.003$ (right).
• Figure 3: Multifidelity: steady state. Expected value of the kinetic density $\mathbb{E}[f]$ for $\varepsilon=0.003$ at different times. Solution computed by Monte Carlo (yellow dashed line), Bi Fidelity (BF) approach with low fidelity the steady state and $\lambda=1$ (blue dashed dotted line) $\lambda=\lambda^*$ (red dashed dotted line), steady state (green dotted line) and reference solution (black dashed dotted line), at times $T=\frac{1}{4}T_f$ (left) and $T=T_f$ (right), for $z \sim \mathcal{U}([1,3])$.
• Figure 4: Multifidelity: steady state. Convergence rate for $\varepsilon=0.003$ of the Monte Carlo method (blue line) and the Bi Fidelity approach (red line) with respect to the stochastic number of samples $M$ for a fixed number of samples in the physical space $N=10^4$ (left), $N=10^5$ (right).
• Figure 5: Multifidelity: BGK model. Expected value of the kinetic density $\mathbb{E}[f]$ for $\varepsilon=0.003$ at different times. Solution computed by Monte Carlo (yellow dashed line), Bi Fidelity (BF) approach with low fidelity BGL approximation and $\lambda=1$ (blue dashed dotted line) $\lambda=\lambda^*$ (red dashed dotted line), steady state (green dotted line) and reference solution (black dashed dotted line), at times $T=\frac{1}{4}T_f$ (left) and $T=T_f$ (right), for $z \sim \mathcal{U}([1,3])$.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• remark 1