Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits...

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Published in:	Journal of scientific computing Vol. 88; no. 2; p. 36
Main Authors:	Dektor, Alec, Rodgers, Abram, Venturi, Daniele
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2021 Springer Nature B.V
Subjects:	Adaptive algorithms Algorithms Approximation Computational Mathematics and Numerical Analysis Covariance matrix Decomposition Energy Fokker-Planck equation Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear differential equations Numerical integration Operators (mathematics) Partial differential equations Robustness (mathematics) Series expansion Tensors Theoretical Time integration Dynamical low-rank approximation Tensor train decomposition Tensor manifolds
ISSN:	0885-7474, 1573-7691
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Abstract	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.
AbstractList	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.
ArticleNumber	36
Author	Venturi, Daniele Dektor, Alec Rodgers, Abram
Author_xml	– sequence: 1 givenname: Alec surname: Dektor fullname: Dektor, Alec organization: Department of Applied Mathematics, University of California Santa Cruz – sequence: 2 givenname: Abram surname: Rodgers fullname: Rodgers, Abram organization: Department of Applied Mathematics, University of California Santa Cruz – sequence: 3 givenname: Daniele orcidid: 0000-0001-8831-8547 surname: Venturi fullname: Venturi, Daniele email: venturi@ucsc.edu organization: Department of Applied Mathematics, University of California Santa Cruz
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Snippet	We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train...
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SubjectTerms	Adaptive algorithms Algorithms Approximation Computational Mathematics and Numerical Analysis Covariance matrix Decomposition Energy Fokker-Planck equation Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear differential equations Numerical integration Operators (mathematics) Partial differential equations Robustness (mathematics) Series expansion Tensors Theoretical Time integration
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Title	Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs
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