# Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties

## TL;DR

This paper develops Monte Carlo gPC numerical schemes for kinetic flocking models with uncertainties, effectively capturing phase transitions and noise effects while preserving positivity and achieving high accuracy.

## Contribution

It introduces a novel combination of Monte Carlo and stochastic Galerkin methods for uncertain kinetic flocking models, ensuring positivity and high accuracy.

## Key findings

- Validated methods on kinetic alignment models with noise.
- Demonstrated accurate capture of phase transition phenomena.
- Showed influence of uncertainties on smoothing and confidence bands.

## Abstract

In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence band.