Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

JOURNAL OF COMPUTATIONAL PHYSICS

卷:404

Article

Dektor, Alec¹  Venturi, Daniele¹ 

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

关键词: High-dimensional PDEs;    Hierarchical tensor methods;    Dynamically orthogonal modes;    Bi-orthogonal decomposition;   

DOI  :  10.1016/j.jcp.2019.109125

来源: Elsevier

【 摘 要 】

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of the binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free