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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  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 & 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.   Google Scholar

[2]

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

[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.  Google Scholar

[4]

M. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.  Google Scholar

[5]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

O. P. L. Maître and O. M. Kino, Spectral Methods for Uncertainty Quantification, Springer, 2010.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[4]

M. Giles, Multilevel Monte Carlo methods, Acta Numerica, 24 (2015), 259-328.  doi: 10.1017/S096249291500001X.  Google Scholar

[5]

P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

O. P. L. Maître and O. M. Kino, Spectral Methods for Uncertainty Quantification, Springer, 2010.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

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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