Particle based gPC methods for mean-field models of swarming with uncertainty
This work addresses the challenge of preserving nonnegativity in stochastic mean-field simulations, which is important for modeling collective behavior but the approach is incremental.
The authors developed numerical schemes for stochastic mean-field equations that preserve nonnegativity by combining Monte Carlo with gPC expansion, achieving high accuracy without losing positivity in applications to swarming models.
In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.