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January  2019, 24(1): 257-272. doi: 10.3934/dcdsb.2018107

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

Mahadevan Ganesh 1, , Brandon C. Reyes 1,, and  Avi Purkayastha 2,

1. 

Department of Applied Mathematics & Statistics, Colorado School of Mines, Golden, CO 80401, USA
2. 

National Renewable Energy Laboratory, Golden, CO 80401, USA

* Corresponding author: B. Reyes

Received  April 2017 Revised  January 2018 Published  January 2019 Early access  March 2018

Fund Project: The authors are supported by an NREL grant UGA-0-41025-117.

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.

Keywords: Diffraction in time, moving-mesh, finite element method, multilevel Monte Carlo, Monte Carlo.
Mathematics Subject Classification: Primary: 35Q41, 65C30, 78M10; Secondary: 65C20.
Citation: Mahadevan Ganesh, Brandon C. Reyes, Avi Purkayastha. An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 257-272. doi: 10.3934/dcdsb.2018107
References:
[1]

C. J. Budd, W. Huang and R. D. Russell, Adaptivity with moving grids, Acta Numerica, 18 (2009), 111-241. 

[2]

A. del Campo, G. Garcia-Calderón and J. Muga, Quantum transients, Phys. Reports, 476 (2009), 1-50. 

[3]

J. Dick, F. Y. Kuo, Q. 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.

[6]

A. Goussev, Diffraction in time: An exactly solvable model, Phys. Review A, 87 (2012), 053621. doi: 10.1103/PhysRevA.87.053621.

[7]

T. Kimura, N. 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. Lee, M. 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.

[10]

M. Moshinsky, Diffraction in time, Phys. Review, 88 (1952), 625-631.  doi: 10.1103/PhysRev.88.625.

[11]

A. Nissen, G. Kreiss and M. Gerritsen, High order stable finite difference methods for the Schrödinger equation, J. Sci. Computing, 55 (2013), 173-199.  doi: 10.1007/s10915-012-9628-1.

[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.

show all references

References:
[1]

C. J. Budd, W. Huang and R. D. Russell, Adaptivity with moving grids, Acta Numerica, 18 (2009), 111-241. 

[2]

A. del Campo, G. Garcia-Calderón and J. Muga, Quantum transients, Phys. Reports, 476 (2009), 1-50. 

[3]

J. Dick, F. Y. Kuo, Q. 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.

[6]

A. Goussev, Diffraction in time: An exactly solvable model, Phys. Review A, 87 (2012), 053621. doi: 10.1103/PhysRevA.87.053621.

[7]

T. Kimura, N. 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. Lee, M. 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.

[10]

M. Moshinsky, Diffraction in time, Phys. Review, 88 (1952), 625-631.  doi: 10.1103/PhysRev.88.625.

[11]

A. Nissen, G. Kreiss and M. Gerritsen, High order stable finite difference methods for the Schrödinger equation, J. Sci. Computing, 55 (2013), 173-199.  doi: 10.1007/s10915-012-9628-1.

[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.

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 $\gamma (t)$.
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Figure 2.  Two realizations of moving-mesh configurations.
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Figure 3.  Simulated density profiles at three discrete times steps
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Figure 6.  Total CPU time to simulate the expected value of the QoI using the MC and MLMC algorithms with $\epsilon = [N_{MC}]^{-1/2}$.
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Figure 4.  Variance for four levels of $Q^{(\ell)}$ and $Q^{(\ell)} - Q^{(\ell-1)}$
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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.
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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
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$N_h$ $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
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
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$N_{MC}$ $10,000$ $50,000$ $100,000$ $500,000$
$\epsilon = [N_{MC}]^{-1/2}$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
$\mathbb{E}_{\rm{mc}} \left[Q; N_{MC} \right]$ 4.6156 4.6199 4.6282 4.6289
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
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$\epsilon$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
$\mathbb{E}_{{\rm{MLMC}}}^{(3)}\left[Q^{(3)}\right]$ 4.6256 4.6307 4.6276 4.6276
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