[Paper Review] Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties

Andrea Medaglia, Andrea Tosin+1 more|arXiv (Cornell University)|Jan 1, 2022

Probabilistic and Robust Engineering DesignDecision Sciences51 references9 citations

TL;DR

This paper introduces a hybrid DSMC-spectral Galerkin method for non-Maxwellian kinetic models of multiagent systems with uncertainties, combining Direct Simulation Monte Carlo in physical space with stochastic Galerkin methods in random space. The approach preserves key physical properties like positivity and entropy dissipation while achieving spectral convergence in the random space through regularization of step functions and correction of nonconserved quantities via surrogate Fokker-Planck models.

ABSTRACT

In this paper, we focus on the construction of a hybrid scheme for the approximation of non-Maxwellian kinetic models with uncertainties. In the context of multiagent systems, the introduction of a kernel at the kinetic level is useful to avoid unphysical interactions. The methods here proposed, combine a direct simulation Monte Carlo (DSMC) in the phase space together with stochastic Galerkin (sG) methods in the random space. The developed schemes preserve the main physical properties of the solution together with accuracy in the random space. The consistency of the methods is tested with respect to surrogate Fokker-Planck models that can be obtained in the quasi-invariant regime of parameters. Several applications of the schemes to non-Maxwellian models of multiagent systems are reported.

Motivation & Objective

• To develop a numerical framework for non-Maxwellian kinetic models with uncertain parameters in multiagent systems.
• To address the challenge of maintaining physical properties (positivity, entropy dissipation) in uncertainty-quantified kinetic simulations.
• To achieve spectral convergence in the random space despite the nonlinearity and non-Maxwellian nature of the interaction kernels.
• To construct a surrogate Fokker-Planck model for quasi-invariant regime approximation to validate the accuracy of the hybrid scheme.
• To ensure robustness and consistency of the method through regularization of step functions and correction of nonconserved quantities.

Proposed method

• Combines Direct Simulation Monte Carlo (DSMC) for particle-level dynamics in physical space with stochastic Galerkin (sG) methods in random space.
• Employs generalized Polynomial Chaos (gPC) expansion to represent the random dependence of DSMC samples.
• Introduces regularization (mollification) of step functions at the particle level to restore spectral convergence in sG.
• Uses a surrogate Fokker-Planck model derived in the quasi-invariant limit to correct nonconserved quantities like mean velocity and energy.
• Applies a two-step correction: regularization of collision rules and rescaling of moments using the Fokker-Planck approximation.
• Employs consistency estimates and convergence studies to validate the method’s accuracy and stability.

Experimental results

Research questions

• RQ1 Can a hybrid DSMC-sG method preserve physical properties (positivity, entropy dissipation) in non-Maxwellian kinetic models with uncertainties?
• RQ2 How can spectral convergence in the random space be restored when step functions break convergence in non-Maxwellian models?
• RQ3 To what extent does the surrogate Fokker-Planck model accurately approximate the moments of the kinetic solution in the quasi-invariant regime?
• RQ4 How do regularization and moment correction affect the accuracy and stability of the DSMC-sG scheme in multiagent systems?
• RQ5 Can the method achieve high accuracy and efficiency in complex models such as wealth distribution and traffic flow with uncertain interactions?

Key findings

• The DSMC-sG method achieves spectral convergence in the random space when combined with step function regularization and moment correction via the surrogate Fokker-Planck model.
• In the gambling model, the method shows L2 error convergence with order ≈5.0 for M1 = M2 = 50, confirming spectral accuracy.
• For the wealth distribution model, the method accurately captures the transition from Pareto-like to Gaussian-like equilibria under uncertainty.
• In the traffic flow model, the method with regularization and rescaling reproduces the expected behavior of mean velocity and energy, with convergence order ≈4.5 for M1 = M2 = 5.
• The surrogate Fokker-Planck model enables accurate approximation of equilibrium distributions and moment evolution, even when the true equilibrium is unknown.
• The method maintains physical consistency (e.g., positivity, bounded moments) across all test cases, even with high-dimensional uncertainty.