POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

Li, Kun; Huang, Ting‐Zhu; Li, Liang; Lanteri, Stéphane

doi:10.1016/j.jcp.2019.05.051

Cited by 22 publications

(12 citation statements)

References 48 publications

(92 reference statements)

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“…where M is the symmetric positive definite matrix, K and S i are the symmetric matrices, and S e and S h are the skew-symmetric matrices. For detailed descriptions of these matrices see [16].…”

Section: Mathematical Modelingmentioning

confidence: 99%

“…During the offline stage, the time-and parameter-independent RB functions are extracted from the collection of full-order solutions (snapshots) generated by the DGTD method at some different parameter locations. There are popular methods comprising the error estimator/indicatorbased greedy algorithm [9,10,11] and the singular value decomposition (SVD)-based proper orthogonal decomposition (POD) method [4,12,13,14,15,16] to generate the RB functions. In particular, the greedy approach is not feasible without a natural criteria for the generation of RB functions [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…According to the method of computing the reduced-order coefficients, the RB methods can be classified into two categories [2,18]: intrusive and non-intrusive. In the former, the ROM is firstly established by using Galerkin projection [4,19,20,21,22] or other means [23,24], which yields a finite-dimensional dynamical system with the smallest possible DoF [16,25,26], and then the reduced-order coefficients are calculated by solving the ROM with the time scheme (e.g., the second order leap-frog (LF 2 ) scheme). MOR for FDTD [27,28], DGTD [16,26], and hybridizable discontinuous Galerkin (HDG) [4,29] methods based on the projection technique has proved to be a powerful method to save computational cost for the PDEs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Huang

et al. 2021

J Sci Comput

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This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract timeand parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI 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 POD-CSI method.

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Section: Mathematical Modelingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Huang

et al. 2021

J Sci Comput

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“…Whether the reduced-order Galerkin methods in [11,26,27], the reduced-order FE methods in [15,17], the reduced-order FVE methods in [18,19], the reduced-order CS methods in [20,21], the reduced-order NBE methods in [22,23] and the reduced basis methods in [24,25] are constructed by the continuous POD basic functions. The constructing process of the continuous POD basic functions requires the knowledge of the optimization methods and functional analysis.…”

Section: Introductionmentioning

confidence: 99%

“…The above reduced-order methods are established by replacing the basic functions in the classical numerical methods (Galerkin, CS, FE, FVE, NBE) with the few continuous POD basic functions, resulting in that these reduced-order methods would emerge with large errors. Moreover, the reduced-order Galerkin methods in [11,26,27] and the reduced-order CS method in [20,21] can only solve the problems defined on rectangular domains.…”

Section: Introductionmentioning

confidence: 99%

The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation

Luo

2020

Mathematics

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This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.

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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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POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

Cited by 22 publications

References 48 publications

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation

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

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