Multiscale Computations for Kinetic and Related Problems

The Vlasov Fokker Planck Equation with High Dimensional Parametric Forcing Term

Yuhua Zhu

University of Wisconsin-Madison

We consider the Vlasov Fokker Planck equation with random electric field, where the random field is paramatrized by a countably infinite random variables . In the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [Cohen-DeVore-15], we prove the best N approximation in the random space breaks the dimensionality 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-14], we develop a residual based adaptive sparse polynomial inter- polation (RASPI) method which is more efficient for dimensionless linear kinetic equation, whose numerical scheme is time dependent and implicit. Although the convergence rate of the RASPI has not been rigorously showed in the paper, but the numerical experiments suggest that it decays independent of dimensionality.