Numerical methods for high-dimensional probability density function equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:305

Article

Cho, H.¹  Venturi, D.³  Karniadakis, G. E.² 

[1] Univ Maryland, Dept Math, College Pk, MD 20742 USA

[2] Brown Univ, Div Appl Math, Providence, RI 02912 USA

[3] Univ Calif Santa Cruz, Dept Appl Math & Stat, Santa Cruz, CA 95064 USA

关键词: High-order numerical methods;    Proper generalized decomposition;    Uncertainty quantification;    Stochastic dynamical systems;    Kinetic partial differential equations;    ANOVA decomposition;   

DOI  :  10.1016/j.jcp.2015.10.030

来源: Elsevier

【 摘 要 】

In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free