Numerical methods for high-dimensional probability density function equations

8 years 14 days ago

Download www.dam.brown.edu

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), systems of stochastic ODEs (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 conditional moments closures and it yields a hierarchy of coupled probability density function equations that resembles the Bogoliubov-BornGreen-Kirkwood-Yvon (BBGKY) hierarchy of kinetic ...

H. Cho, D. Venturi, George E. Karniadakis

Real-time Traffic