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

Kazashi, Yoshihito; Nobile, Fabio; Vidličková, Eva

doi:10.1007/s00211-021-01241-4

Cited by 10 publications

(4 citation statements)

References 39 publications

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“…The projector-splitting schemes proposed by Lubich and Oseledets [ 24 ] were the first to remedy this issue. Since then, a collection of methods have been designed that achieve stability independently of the presence of small singular values [ 7 – 9 , 17 , 18 ]. As we show in the following section, some of these DLRA algorithms are directly related to retractions.…”

Section: Preliminariesmentioning

confidence: 99%

From low-rank retractions to dynamical low-rank approximation and back

Séguin,

Ceruti,

Kressner

2024

Bit Numer Math

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In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

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Section: Preliminariesmentioning

confidence: 99%

From low-rank retractions to dynamical low-rank approximation and back

Séguin,

Ceruti,

Kressner

2024

Bit Numer Math

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“…Lubich (2018, 2019) constructed projectorsplitting integrators for Vlasov-Poisson equations. For random parabolic equations, a scheme of this type is analysed in Kazashi, Nobile and Vidličková (2021). Variants including strategies for the adaptation of approximation ranks are proposed in Yang and White (2020), Dektor, Rodgers and Venturi (2021) and Dunnett and Chin (2021).…”

Section: Splitting and Basis Update And Galerkin Integratorsmentioning

confidence: 99%

Low-rank tensor methods for partial differential equations

Bachmayr¹

2023

Acta Numerica

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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“…In recent years, there has been a growing interest in the development of MOR techniques for parametric dynamical systems to overcome the limitations of linear global approximations. A large class of methods consider the dynamical low rank (DLR) approximation (see [17][18][19][20][21]) which allows both the deterministic and stochastic basis functions to evolve in time. Other strategies based on deep learning (DL) algorithms were proposed in [22][23][24] to construct the efficient surrogate model for time-dependent parametrized PDEs.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems

Li,

Liu,

null

et al. 2024

CiCP

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Parametric dynamical systems are widely used to model physical systems, but their numerical simulation can be computationally demanding due to nonlinearity, long-time simulation, and multi-query requirements. Model reduction methods aim to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with nonlinear structures. To address this challenge, we propose an adaptive method based on local dynamic mode decomposition (DMD) to construct an efficient and reliable surrogate model. We propose an improved greedy algorithm to generate the atoms set Θ based on a sequence of relatively small training sets, which could reduce the effect of large training set. At each enrichment step, we construct a local sub-surrogate model using the Taylor expansion and DMD, resulting in the ability to predict the state at any time without solving the original dynamical system. Moreover, our method provides the best approximation almost everywhere over the parameter domain with certain smoothness assumptions, thanks to the gradient information. At last, three concrete examples are presented to illustrate the effectiveness of the proposed method.

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

Cited by 10 publications

References 39 publications

From low-rank retractions to dynamical low-rank approximation and back

From low-rank retractions to dynamical low-rank approximation and back

Low-rank tensor methods for partial differential equations

An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems

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