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

In traffic flow modelling, 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 vehicle 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.