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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Vydané v:Journal of scientific computing Ročník 88; číslo 2; s. 36
Hlavní autori: Dektor, Alec, Rodgers, Abram, Venturi, Daniele
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.08.2021
Springer Nature B.V
Predmet:
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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Popis
Shrnutí: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.
Bibliografia:ObjectType-Article-1
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-021-01539-3