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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

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.

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

[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.

[6]

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

[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.

[10]

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

[11]

A. NissenG. 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. BuddW. Huang and R. D. Russell, Adaptivity with moving grids, Acta Numerica, 18 (2009), 111-241. 

[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.

[6]

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

[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.

[10]

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

[11]

A. NissenG. 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 $\gamma (t)$.
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
$N_h$ $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
$N_{MC}$ $10,000$ $50,000$ $100,000$ $500,000$
$\epsilon = [N_{MC}]^{-1/2}$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
$\mathbb{E}_{\rm{mc}} \left[Q; N_{MC} \right]$ 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
$\epsilon$ $0.0100$ $0.0045$ $0.0032$ $0.0014$
$\mathbb{E}_{{\rm{MLMC}}}^{(3)}\left[Q^{(3)}\right]$ 4.6256 4.6307 4.6276 4.6276
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