The Vlasov-Fokker-Planck equation with high dimensional parametric forcing term

Jin S, Zhu Y, Zuazua E (2022)

Publication Type: Journal article

Publication year: 2022

Journal

Numerische Mathematik Springer Verlag (Germany)

DOI: 10.1007/s00211-021-01257-w

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 and DeVore in Acta Numerica 24:1-159, 2015) , we prove the best N approximation in the random space enjoys a convergence rate, which depends on the summability of the coefficients of the random variable, higher than the Monte-Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in Chkifa et al. (Found Comput Math 14:601-603, 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.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Shanghai Jiao Tong University / 上海交通大学 CN China (CN) University of Wisconsin - Madison US United States (USA) (US)

How to cite

APA:

Jin, S., Zhu, Y., & Zuazua, E. (2022). The Vlasov-Fokker-Planck equation with high dimensional parametric forcing term. Numerische Mathematik. https://dx.doi.org/10.1007/s00211-021-01257-w

MLA:

Jin, Shi, Yuhua Zhu, and Enrique Zuazua. "The Vlasov-Fokker-Planck equation with high dimensional parametric forcing term." Numerische Mathematik (2022).

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