Dynamic tensor approximation of high-dimensional nonlinear PDEs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:437

Article

Dektor, Alec¹  Venturi, Daniele¹ 

[1] Univ Calif Santa Cruz, Dept Appl Math, Santa Cruz, CA 95064 USA

关键词: Hierarchical tensor methods;    Functional tensor train;    Fokker-Planck equation;   

DOI  :  10.1016/j.jcp.2021.110295

来源: Elsevier

【 摘 要 】

We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This minimization problem yields the unique optimal velocity vector that allows us to integrate the PDE forward in time on a tensor manifold of constant rank. In the case of initial/boundary value problems defined in real separable Hilbert spaces, this procedure yields evolution equations for the tensor modes in the form of a coupled system of one-dimensional time-dependent PDEs. We apply the dynamic tensor approximation to a four-dimensional Fokker-Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free