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

  • *Corresponding author: Ting-Zhu Huang, Liang Li

    *Corresponding author: Ting-Zhu Huang, Liang Li 

The first author is supported by the NSFC (Grant No. 12101511). The second author is supported by the NSFC (Grant No. 61772003) and the Key Projects of Applied Basic Research in Sichuan Province (Grant No. 2020YJ0216)

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  • 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 $ L $ ($ L\geq1 $) 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 $ L+1 $, which includes the sparse grids from approximation level $ L $. 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.

    Mathematics Subject Classification: Primary: 78M34; Secondary: 65M22.

    Citation:

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  • Figure 1.  1-D Smolyak sparse grids with approximation levels 0, 1, 2, 3

    Figure 2.  2-D Smolyak sparse grid with approximation level 3 and full tensor product grid

    Figure 3.  Scattering of plane wave by a dielectric disk: the convergence histories of $ \overline{e}_{ {\bf E}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}^{(i)}} $ (a), $ \overline{e}_{ {\bf H}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}^{(i)}} $ (b) ($ i = 1, 2 $) on the testing set $ \mathcal{T}_{te}\times\mathcal{P}_{te} $ with vary truncation tolerances $ \rho_\theta $, where $ \overline{e}_{ {\bf u}, {\rm{Pro}}^{(i)}} $ is the average projection error of $ {\rm{NIMOR}}^{(i)} $ method for $ {\bf u} $

    Figure 4.  Scattering of plane wave by a dielectric disk: the $ 5 $-th, $ 10 $-th, $ 15 $-th, and $ 20 $-th exact and approximation reduced-order cofficients of $ E_z $ based on $ {\rm{NIMOR}}^{(1)} $

    Figure 5.  Scattering of plane wave by a dielectric disk: the $ 5 $-th, $ 10 $-th, $ 15 $-th, and $ 20 $-th exact and approximation reduced-order cofficients of $ H_y $ based on $ {\rm{NIMOR}}^{(1)} $

    Figure 6.  Scattering of plane wave by a dielectric disk: comparison of the 1-D x-wise distribution along $ y = 0 $ of the real part of $ H_y $ (left) and $ E_z $ (right) of four test points: $ \theta^{(1)} = 1.215 $ (1-th row), $ \theta^{(2)} = 2.215 $ (2-th row), $ \theta^{(3)} = 3.215 $ (3-th row) and $ \theta^{(4)} = 4.215 $ (4-th row)

    Figure 7.  Scattering of plane wave by a dielectric disk: comparison of relative projection and NIMOR errors for $ {\bf E} $ (left) and $ {\bf H} $ (right) four the testing parameters

    Figure 8.  Scattering of a plane wave by a multi-layer heterogeneous medium: geometry of the multi-layer medium

    Figure 9.  Scattering of a plane wave by a multi-layer heterogeneous medium: the convergence histories of $ \overline{e}_{ {\bf E}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}^{(i)}} $ (a), $ \overline{e}_{ {\bf H}, {\rm{Pro}}^{(i)}} $, and $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}^{(i)}} $ (b) ($ i = 1, 2 $) on the testing set $ \mathcal{T}_{te}\times\mathcal{P}_{te} $ with vary truncation tolerances $ \rho_\theta $, where $ \overline{e}_{ {\bf u}, {\rm{Pro}}^{(i)}} $ is the average projection error of $ {\rm{NIMOR}}^{(i)} $ method for $ {\bf u} $

    Figure 10.  Scattering of plane wave by a multi-layer heterogeneous medium: the 3th, 6th, 9th, and 12th exact ($ - $) and approximation ($ * $) time-modes for $ E_z $ (left) and $ H_y $ (right) (the 1th mode: black, the 3th mode: red, the 5th mode: brown; the 7th mode: blue; the 9th mode: green)

    Figure 11.  Scattering of plane wave by a multi-layer heterogeneous medium: comparison of the 1-D x-wise distribution along $ y = 0 $ of the real part of $ H_y $ (left) and $ E_z $ (right) of four test points: $ \theta^{(1)} = \{(5.125, 3.375, 2.125, 1.375)\} $ (1-th row), $ \theta^{(2)} = \{(5.425, 3.625, 2.425, 1.625)\} $ (2-th row), $ \theta^{(3)} = \{(5.125, 3.625, 2.125, 1.625)\} $ (3-th row) and $ \theta^{(4)} = \{(5.425, 3.375, 2.425, 1.375)\} $ (4-th row)

    Figure 12.  Scattering of plane wave by a multi-layer heterogeneous medium: comparison of relative projection and NIMOR errors for $ {\bf E} $ (left) and $ {\bf H} $ (right) four the testing parameters

    Table 1.  Comparison of the number of sampling points using Smolyak sparse gird and full tensor product grid. Here $ M_1 $ is the number of sampling points using Smolyak sparse gird, and $ M_2 $ is the number of sampling points using full tensor product grid

    Dimension size $ p $ Approximation level $ l $ $ M_1 $ $ M_2 $ $ \dfrac{M_2}{M_1} $
    2 13 25 1.923
    2 3 29 81 2.793
    5 145 1089 7.510
    2 61 3125 51.230
    5 3 241 59049 $ 2.450\times10^2 $
    5 2433 $ 3.914\times10^7 $ $ 1.609\times10^4 $
    2 221 $ 9.766\times10^6 $ $ 4.419\times10^4 $
    10 3 1581 $ 3.487\times10^9 $ $ 2.206\times10^6 $
    5 41265 $ 1.532\times10^{15} $ $ 3.713\times10^{10} $
     | Show Table
    DownLoad: CSV

    Table 2.  Scattering of plane wave by a dielectric disk: settings for the training, and testing datasets

    Data set Training set Testing set
    Parameter sample points 65, uneven (Smolyak mehod) 40, random (LHS method)
    Time sample points 263, uniform 263, uniform
    Size 17095 10520
     | Show Table
    DownLoad: CSV

    Table 3.  Scattering of plane wave by a dielectric disk: the average projection and NIMOR errors on the testing set

    Average relative errors $ \overline{e}_{ {\bf E}, {\rm{Pro}}} $ $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}} $ $ \overline{e}_{ {\bf H}, {\rm{Pro}}} $ $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}} $
    $ {\rm{NIMOR}}^{(1)} {\rm{method}} $ $ 1.170\times 10^{-2} $ $ 1.500\times 10^{-2} $ $ 1.065\times 10^{-2} $ $ 1.447\times 10^{-2} $
    $ {\rm{NIMOR}}^{(2)} {\rm{method}} $ $ 1.173\times 10^{-2} $ $ 1.279\times 10^{-2} $ $ 1.069\times 10^{-2} $ $ 1.198\times 10^{-2} $
     | Show Table
    DownLoad: CSV

    Table 4.  Scattering of plane wave by a dielectric disk: computational times of $ {\rm{NIMOR}}^{(i)} $ ($ i = 1, 2 $) and DGTD methods in terms of CPU time. The unit of time cost is second

    Method Offlin stage
    (Snapshots, Nested POD, GRP training)
    Online stage
    (one run for new paramter)
    DGTD - $ 4.254\times10^{2} $
    $ {\rm{NIMOR}}^{(1)} $ $ 1.444\times10^{4} $ 3.8
    $ {\rm{NIMOR}}^{(2)} $ $ 2.793\times10^{4} $ 3.1
     | Show Table
    DownLoad: CSV

    Table 5.  Scattering of a plane wave by a multi-layer heterogeneous medium: the distribution and range of material parameters

    Layer $ i $ $ \mathcal{P}^{(i)} $ $ \mu_{r, i} $ $ r_i $
    1 $ \varepsilon_{r, 1}\in[5.0, 5.6] $ 1 0.15
    2 $ \varepsilon_{r, 2}\in[3.25, 3.75] $ 1 0.3
    3 $ \varepsilon_{r, 3}\in[2.0, 2.5] $ 1 0.45
    4 $ \varepsilon_{r, 4}\in[1.25, 1.75] $ 1 0.6
     | Show Table
    DownLoad: CSV

    Table 6.  Scattering of plane wave by a multi-layer heterogeneous medium: settings for the training, and testing datasets

    Data set Training set Testing set
    Parameter sample points 137, uneven (Smolyak mehod) 40, random (LHS method)
    Time sample points 253, uniform 253, uniform
    Size 34661 10120
     | Show Table
    DownLoad: CSV

    Table 7.  Scattering of plane wave by a multi-layer heterogeneous medium: the average projection and NIMOR errors on the testing set

    Average relative errors $ \overline{e}_{ {\bf E}, {\rm{Pro}}} $ $ \overline{e}_{ {\bf E}, {\rm{NIMOR}}} $ $ \overline{e}_{ {\bf H}, {\rm{Pro}}} $ $ \overline{e}_{ {\bf H}, {\rm{NIMOR}}} $
    $ {\rm{NIMOR}}^{(1)} {\rm{method}} $ $ 3.668\times 10^{-3} $ $ 6.011\times 10^{-3} $ $ 4.419\times 10^{-3} $ $ 6.695\times 10^{-3} $
    $ {\rm{NIMOR}}^{(2)} {\rm{method}} $ $ 3.665\times 10^{-3} $ $ 4.744\times 10^{-3} $ $ 4.405\times 10^{-3} $ $ 5.229\times 10^{-3} $
     | Show Table
    DownLoad: CSV

    Table 8.  Scattering of plane wave by a multi-layer heterogeneous medium: computational times of $ {\rm{NIMOR}}^{(i)} $ ($ i = 1, 2 $) and DGTD methods in terms of CPU time. The unit of time cost is second

    Method Offlin stage
    (Snapshots, Nested POD, GRP training)
    Online stage
    (one run for new paramter)
    DGTD - $ 4.513\times10^{2} $
    $ {\rm{NIMOR}}^{(1)} $ $ 1.912\times10^{4} $ 3.9
    $ {\rm{NIMOR}}^{(2)} $ $ 4.221\times10^{4} $ 3.3
     | Show Table
    DownLoad: CSV
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