Tensor methods for the Boltzmann-BGK equation

Dimension reduction is analytical methods for reconstructing high-order tensors that the intrinsic rank of these tensor data is relatively much smaller than the dimension of the ambient measurement space. Typically, this is the case for most real world datasets in signals, images and machine learning. The CANDECOMP/PARAFAC (CP, aka Canonical Polyadic) tensor completion is a widely used approach to find a low-rank approximation for a given tensor. In the tensor model (Sanogo and Navasca in 2018 52nd Asilomar conference on signals, systems, and computers, pp 845–849, 10.1109/ACSSC.2018.8645405, 2018), a sparse regularization minimization problem via $$\ell _1$$ ℓ 1 norm was formulated with an appropriate choice of the regularization parameter. The choice of the regularization parameter is important in the approximation accuracy. Due to the emergence of the massive data, one is faced with an onerous computational burden for computing the regularization parameter via classical approaches (Gazzola and Sabaté Landman in GAMM-Mitteilungen 43:e202000017, 2020) such as the weighted generalized cross validation (WGCV) (Chung et al. in Electr Trans Numer Anal 28:2008, 2008), the unbiased predictive risk estimator (Stein in Ann Stat 9:1135–1151, 1981; Vogel in Computational methods for inverse problems, 2002), and the discrepancy principle (Morozov in Doklady Akademii Nauk, Russian Academy of Sciences, pp 510–512, 1966). In order to improve the efficiency of choosing the regularization parameter and leverage the accuracy of the CP tensor, we propose a new algorithm for tensor completion by embedding the flexible hybrid method (Gazzola in Flexible krylov methods for lp regularization) into the framework of the CP tensor. The main benefits of this method include incorporating the regularization automatically and efficiently as well as improving accuracy in the reconstruction and algorithmic robustness. Numerical examples from image reconstruction and model order reduction demonstrate the efficacy of the proposed algorithm.

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“…In addition, tensor based methods are gaining grounds in solving complex problems in scientific computing. [11].…”

Section: Introductionmentioning

confidence: 99%

Low-CP-Rank Tensor Completion via Practical Regularization

2022

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“…High-dimensional partial differential equations (PDEs) arise in many areas of engineering, physical sciences and mathematics. Classical examples are equations involving probability density functions (PDFs) such as the Fokker-Planck equation [49], the Liouville equation [13,14,58], and the Boltzmann equation [8,11,18]. More recently, high-dimensional PDEs have also become central to many new areas of application such as optimal mass transport [27,59], random dynamical systems [57,58], mean field games [19,52], and functionaldifferential equations [55,56].…”

Section: Introductionmentioning

confidence: 99%

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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“…Effective methods to represent such functional dependence are based on polynomial chaos expansions [34,[97][98][99]102], probabilistic collocation methods [29,36,101], and deep neural networks [74,104]. Other techniques rely on a reformulation of the problem in terms of kinetic equations [11,19,96], or hierarchies of kinetic equations [13,18,94].…”

Section: Introductionmentioning

confidence: 99%

Spectral methods for nonlinear functionals and functional differential equations

Venturi

Dektor

2021

Res Math Sci

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We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.

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Tensor methods for the Boltzmann-BGK equation

Cited by 21 publications

References 50 publications

Low-CP-Rank Tensor Completion via Practical Regularization

Low-CP-Rank Tensor Completion via Practical Regularization

Rank-Adaptive Tensor Methods for High-Dimensional Nonlinear PDEs

Spectral methods for nonlinear functionals and functional differential equations

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