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An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model

  • * Corresponding author: B. Reyes

    * Corresponding author: B. Reyes 
The authors are supported by an NREL grant UGA-0-41025-117.
Abstract Full Text(HTML) Figure(6) / Table(3) Related Papers Cited by
  • We consider a moving shutter and non-deterministic generalization of the diffraction in time model introduced by Moshinsky several decades ago to study a class of quantum transients. We first develop a moving-mesh finite element method (FEM) to simulate the determisitic version of the model. We then apply the FEM and multilevel Monte Carlo (MLMC) algorithm to the stochastic moving-domain model for simulation of approximate statistical moments of the density profile of the stochastic transients.

    Mathematics Subject Classification: Primary: 35Q41, 65C30, 78M10; Secondary: 65C20.

    Citation:

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  • Figure 1.  An illustration of the DiT model governed by a moving shutter with speed $\gamma\,^\prime (t)$ and a continuous initial state. The shutter on the left represents the closed shutter's position at $t = 0$ and the right shutter is a closed shutter at the position determined by $\gamma (t)$.

    Figure 2.  Two realizations of moving-mesh configurations.

    Figure 3.  Simulated density profiles at three discrete times steps

    Figure 6.  Total CPU time to simulate the expected value of the QoI using the MC and MLMC algorithms with $\epsilon = [N_{MC}]^{-1/2}$.

    Figure 4.  Variance for four levels of $Q^{(\ell)}$ and $Q^{(\ell)} - Q^{(\ell-1)}$

    Figure 5.  Adaptively chosen values of $N_{{\rm{MLMC}}}^{(\ell)},~\ell = 0,1, 2, 3$ for four distinct choices of $\epsilon$ as in Table 3.

    Table 1.  Convergence of the moving-mesh FEM DiT model using a reference solution with $M_{\Delta t}^{fine} = N_h^{fine} = 20480$.

    $N_h$ $Err^{max}(N_h; 20480) $ EOC of $Err^{max}$
    80 3.1601e-01
    160 1.3607e-01 1.2156
    320 5.3584e-02 1.3445
    640 2.1350e-02 1.3276
    1280 1.0176e-02 1.0691
    2560 4.9566e-03 1.0377
    5120 1.7975e-03 1.4633
    10240 3.8505e-04 2.2229
     | Show Table
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    Table 2.  $\mathbb{E}_{\rm{mc}} \left[Q; N_{MC} \right]$ values obtained using the standard MC method.

    $N_{MC}$ $10,000$ $50,000$ $100,000$ $500,000$
    $\epsilon = [N_{MC}]^{-1/2}$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
    $\mathbb{E}_{\rm{mc}} \left[Q; N_{MC} \right]$ 4.6156 4.6199 4.6282 4.6289
     | Show Table
    DownLoad: CSV

    Table 3.  $\mathbb{E}_{{\rm{MLMC}}}^{(3)}\left[Q^{(3)}\right]$ values obtained using the MLMC algorithm with level dependent space-time mesh parameters as stated in (35).

    $\epsilon$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
    $\mathbb{E}_{{\rm{MLMC}}}^{(3)}\left[Q^{(3)}\right]$ 4.6256 4.6307 4.6276 4.6276
     | Show Table
    DownLoad: CSV
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    [2] A. del CampoG. Garcia-Calderón and J. Muga, Quantum transients, Phys. Reports, 476 (2009), 1-50. 
    [3] J. DickF. Y. KuoQ. L. Gia and C. Schwab, Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations, SIAM J. NUMER. ANAL., 54 (2016), 2541-2568.  doi: 10.1137/16M1078690.
    [4] M. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.
    [5] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.
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    [7] T. KimuraN. Sato and S. Iwata, Application of the higher order finite-element method to one-dimensional Schrödinger equation, J. Comput. Chem., 9 (1998), 827-835.  doi: 10.1002/jcc.540090805.
    [8] T. E. LeeM. J. Baines and S. Langdon, A finite difference moving mesh method based on conservation for moving boundary problems, J. Comput. Appl. Math., 288 (2015), 1-17.  doi: 10.1016/j.cam.2015.03.032.
    [9] O. P. L. Maître and O. M. Kino, Spectral Methods for Uncertainty Quantification, Springer, 2010.
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    [12] Z. Romanowski, Application of h-adaptive, high order finite element method to solve radial Schrödinger equation, Molecular Physics, 107 (2009), 1339-1348.  doi: 10.1080/00268970902873554.
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