Dynamic tensor approximation of high-dimensional nonlinear PDEs

Dektor, Alec; Venturi, Daniele

doi:10.1016/j.jcp.2021.110295

Cited by 17 publications

(26 citation statements)

References 41 publications

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“…It can be shown that ψ i (α i−1 ; x i ; α i ) are eigenfunctions for a self-adjoint compact operator and λ(1) ≥ λ(2) ≥ • • • ≥ 0 is a sequence of real numbers converging to zero (see [6,17] for more details). By truncating (5) so that only the largest singular values are retained, we obtain the approximation of u(x)…”

Section: The Manifold Of Fixed-rank Ftt Tensorsmentioning

confidence: 99%

“…consisting of fixed-rank FTT tensors, is a smooth Hilbert submanifold of L 2 μ ( ) (see [17]). We represent elements in the tangent space, T u r M r , of M r at the point u r ∈ M r as equivalence classes of velocities of continuously differentiable curves on M r passing through u r…”

Section: Tangent and Normal Spaces Of Fixed-rank Ftt Manifoldsmentioning

confidence: 99%

“…The solution to ( 27) is known as a dynamic approximation to the solution of (1). In the context of separable Hilbert spaces, the dynamic approximation problem ( 27) can be solved using dynamically orthogonal or bi-orthogonal constraints on tensor modes [16,17]. Such constraints, also referred to as gauge conditions, provide the unique solution of the minimization problem ( 26) with different FTT cores.…”

Section: Dynamic Tensor Approximation On Low-rank Ftt Manifoldsmentioning

confidence: 99%

“…In this paper, we build upon our recent work on dynamical tensor approximation [16,17], and develop new rank-adaptive temporal integrators to compute the numerical solution of high-dimensional initial/boundary value problems of the form…”

Section: Introductionmentioning

confidence: 99%

“…Hence, to integrate a solution characterized by tensor modes with a wide range of energies one has to consider time stepping schemes that can effectively handle geometric features associated with the curvature of the manifold. In projection-based approaches [16,17,34,43] the computational challenge posed by the curvature of the tensor manifold translates into the need to invert the positive semidefinite covariance matrices of the tensor cores at each time step. A time-integration scheme constructed in this way may become numerically unstable in the presence of tensor modes with small energy, or even singular when modes with zero energy are present (e.g., at a time instant in which we increase the tensor rank by adding a mode with zero energy).…”

Section: Introductionmentioning

confidence: 99%

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Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

2021

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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.

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Section: The Manifold Of Fixed-rank Ftt Tensorsmentioning

confidence: 99%

Section: Tangent and Normal Spaces Of Fixed-rank Ftt Manifoldsmentioning

confidence: 99%

Section: Dynamic Tensor Approximation On Low-rank Ftt Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

2021

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Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

2022

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We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.

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Implicit Integration of Nonlinear Evolution Equations on Tensor Manifolds

Rodgers,

Venturi

2023

J Sci Comput

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Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations. However, the combination of non-linearity and stiffness may introduce time-step restrictions which could make explicit integration computationally infeasible. To overcome this problem, we develop a new class of implicit rank-adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms are based on performing one time step with a conventional time-stepping scheme, followed by an implicit fixed point iteration step involving a rank-adaptive truncation operation onto a tensor manifold. Implicit step truncation methods are straightforward to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Numerical applications demonstrating the effectiveness of implicit step-truncation tensor integrators are presented and discussed for the Allen–Cahn equation, the Fokker–Planck equation, and the nonlinear Schrödinger equation.

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Dynamic tensor approximation of high-dimensional nonlinear PDEs

Cited by 17 publications

References 41 publications

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Implicit Integration of Nonlinear Evolution Equations on Tensor Manifolds

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