The Vlasov-Fokker-Planck Equation with High Dimensional Parametric Forcing Term

TL;DR

This paper studies the Vlasov-Fokker-Planck equation with a high-dimensional random electric field, proving that certain approximation methods can overcome the curse of dimensionality and outperform Monte Carlo in accuracy and efficiency.

Contribution

It introduces a residual-based adaptive sparse polynomial interpolation method for high-dimensional stochastic kinetic equations, demonstrating improved convergence and efficiency over traditional methods.

Findings

RASPI outperforms Monte Carlo in error decay rate

The method is dimension independent

Convergence rate exceeds that of Monte Carlo

Abstract

We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [Cohen, DeVore, 2015], we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in [Chkifa, Cohen, Schwab, 2014], we develop a residual-based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time-dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.