Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Rodgers, Abram; Dektor, Alec; Venturi, Daniele

doi:10.1007/s10915-022-01868-x

Cited by 9 publications

(15 citation statements)

References 96 publications

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“…Integrating the PDE (20) forward in on a FTT tensor manifold, e.g. using rank-adaptive step-truncation methods [32,31,20] or dynamic tensor approximation methods [11,10,9,24,21], results in a FTT approximation v TT (x; ) of the function v(x; ) for all ≥ 0. The computational cost of this approach for computing the FTT expansion of a FTT ridge function is precisely the same cost as solving the hyperbolic PDE (20) in the FTT format, which in the case of step-truncation or dynamic approximation has computational complexity that scales linearly with d. Note that the accuracy of v TT (x; ) as an approximation of v(x; ) depends on the step-size and order of integration scheme used to solve the PDE (20).…”

Section: Computing Tensor Ridge Functions Via Coordinate Flowsmentioning

confidence: 99%

“…It is straightforward to verify that ( 25) satisfies (26). Note that v(x; ) in ( 25) is not an FTT tensor if = πk/2 and k ∈ N. To compute the FTT representation of v TT (x; ) we can solve the PDE (26) on a tensor manifold using step-truncation or dynamic tensor approximation methods [9,11,31,33]. Given the low dimensionality of the spatial domain in this example (d = 2), we can also evaluate (25) directly, and compute its FTT decomposition by solving an eigenvalue problem.…”

Section: Computing Tensor Ridge Functions Via Coordinate Flowsmentioning

confidence: 99%

“…It is convenient to use a step-truncation method [31,33,20] to integrate the initial value problem (45) on a FTT tensor manifold. This is because applying the FTT truncation operation to v(x; ) requires computing the orthogonalized tensor cores Q ≤i ( ), Q >i ( ) (i = 1, 2, .…”

Section: Numerical Integration Of the Gradient Descent Equationsmentioning

confidence: 99%

“…In order to guarantee that the solution v i+1 is a low-rank FTT tensor, we apply a truncation operator to the right hand side of (47). This yields the step-truncation method [31]…”

Section: Numerical Integration Of the Gradient Descent Equationsmentioning

confidence: 99%

“…From this dynamical system perspective we derive a partial differential equation (PDE) for the -dependent function v(x; ) = u(A( )x). We then integrate the PDE for v(x; ) using a step-truncation tensor integration scheme [31,33,20] to obtain v(x; ) in a low-rank tensor format for all . The quasi-optimal rankreducing coordinate map A is obtained by minimizing a rank-related cost function using an appropriate path of transformations A( ).…”

Section: Introductionmentioning

confidence: 99%

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Tensor rank reduction via coordinate flows

Dektor¹,

Venturi²

2022

Preprint

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Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and the numerical solution to highdimensional PDEs. In this paper, we propose a new tensor rank reduction method that leverages coordinate flows and can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is very simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has smaller tensor rank. We restrict our analysis to linear coordinate transformations, which give rise to a new class of functions that we refer to as tensor ridge functions. By leveraging coordinate flows and tensor ridge functions, we develop an optimization method based on Riemannian gradient descent for determining a quasi-optimal linear coordinate transformation for tensor rank reduction. The theoretical results we present for rank reduction via linear coordinate transformations can be generalized to larger classes of nonlinear transformations. We demonstrate the effectiveness of the proposed new tensor rank reduction method on prototype function approximation problems, and in computing the numerical solution of the Liouville equation in dimensions three and five.

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Section: Computing Tensor Ridge Functions Via Coordinate Flowsmentioning

confidence: 99%

Section: Computing Tensor Ridge Functions Via Coordinate Flowsmentioning

confidence: 99%

Section: Numerical Integration Of the Gradient Descent Equationsmentioning

confidence: 99%

“…In order to guarantee that the solution v i+1 is a low-rank FTT tensor, we apply a truncation operator to the right hand side of (47). This yields the step-truncation method [31]…”

Section: Numerical Integration Of the Gradient Descent Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tensor rank reduction via coordinate flows

Dektor¹,

Venturi²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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Low-rank Parareal: a low-rank parallel-in-time integrator

2023

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In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge–Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

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Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations

Sutti,

Vandereycken

2024

J Sci Comput

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We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen–Cahn and the Fisher–KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

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Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Cited by 9 publications

References 96 publications

Tensor rank reduction via coordinate flows

Tensor rank reduction via coordinate flows

Low-rank Parareal: a low-rank parallel-in-time integrator

Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations

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