An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

Xu, Baozou; Zhang, Xiaohua

doi:10.1186/s13662-019-2273-3

Cited by 7 publications

(2 citation statements)

References 27 publications

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“…ROM is created by constructing the reduced subspace composed of a set of low-dimensional POD basis vectors, which are generated by the snapshots, and seeking approximation solutions within this reduced subspace via Galerkin projection or other means. POD-Galerkin method combined with some numerical methods, such as finite difference (FD) [54,55,58], FE [17,21,46], FV [30,43], and DG [24,27] methods, has proved to be a powerful technique to save computational time for the time-domain partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

Huang

et al. 2022

Numerical Methods Partial

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In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF 2 ) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems.

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Section: Introductionmentioning

confidence: 99%

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

Huang

et al. 2022

Numerical Methods Partial

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“…To determine the reduced-order or expansion coefficients of the POD basis functions during the online stage, the MOR methods are generally grouped into two families: intrusive MOR (IMOR) and non-intrusive MOR (NIMOR) methods. In IMOR method, the POD method combined with projection techniques are usually used to reduce the complexity of classical numerical methods, such as the finite difference (FD) [32,54,51], FE [21,43,18], FV [31,26,41], hybridizable discontinuous Galerkin (HDG) [44], and DGTD [24,23] methods, where the Galerkin procedure is the most popular choice for the projection [46]. Besides, in order to avoid the unrewarding repeated computations in the IMOR methods, some reduced-order extrapolated schemes have successively been established by Luo's research team since 2013 [28,29,33,34,36,35,30,55].…”

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confidence: 99%

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

Huang

et al. 2023

DCDS-B

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<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

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A Krylov-based proper orthogonal decomposition method for elastodynamics problems with isogeometric analysis

Liu¹,

Wang²,

Yu³

et al. 2021

Engineering Analysis with Boundary Elements

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An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

Cited by 7 publications

References 27 publications

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

A Krylov-based proper orthogonal decomposition method for elastodynamics problems with isogeometric analysis

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