Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems

Kieri, Emil; Vandereycken, Bart

doi:10.1515/cmam-2018-0029

Cited by 34 publications

(77 citation statements)

References 31 publications

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“…We remark that these a-priori results only hold for (very) short times. In practice, they are overly pessimistic and in actual problems the accuracy is typically much higher than theoretically predicted; see [57,71,72,83,94], and the numerical example from Fig. 9.8 further below.…”

Section: Approximation Propertiesmentioning

confidence: 94%

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Geometric Methods on Low-Rank Matrix and Tensor Manifolds

Uschmajew

Vandereycken

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors.

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Section: Approximation Propertiesmentioning

confidence: 94%

“…9.7. In order to quantity the effect of this approximation on the global error at the final time T , the simplest analysis is to assume as in [57,58] that the vector field F is ε close to the tangent bundle of M, that is,…”

Section: Approximation Propertiesmentioning

confidence: 99%

Geometric Methods on Low-Rank Matrix and Tensor Manifolds

Uschmajew

Vandereycken

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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“…, d − 1, remain orthonormal for all t ≥ 0. We have shown in [17] that under these constraints the convex minimization problem (28) admit a unique minimum for vectors in the tangent space (31) satisfying the PDE systeṁ…”

Section: Dynamic Tensor Approximation On Low-rank Ftt Manifoldsmentioning

confidence: 99%

“…Step-truncation methods Another methodology to integrate nonlinear PDEs on fixed-rank tensor manifolds M r is step-truncation [31,50,49]. The idea is to integrate the solution off of M r for short time, e.g., by performing one time step of the full equation with a conventional time-stepping scheme, followed by a truncation operation back onto M r .…”

Section: 2mentioning

confidence: 99%

“…Computing the numerical solution to high-dimensional PDEs is an extremely challenging problem which has attracted substantial research efforts in recent years. Techniques such as sparse collocation methods [10,13,6,24,39], highdimensional model representations [34,11,5], deep neural networks [45,46,58], and numerical tensor methods [29,4,49,8,27,31] were proposed to mitigate the exponential growth of the degrees of freedom, the computational cost and the memory requirements.…”

Section: Introductionmentioning

confidence: 99%

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Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Dektor

Venturi

2020

Journal of Computational Physics

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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 be 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 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.

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Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

2021

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We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

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Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems

Cited by 34 publications

References 31 publications

Geometric Methods on Low-Rank Matrix and Tensor Manifolds

Geometric Methods on Low-Rank Matrix and Tensor Manifolds

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

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