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

Mahadevan Ganesh; Brandon C. Reyes; Avi Purkayastha

doi:10.3934/dcdsb.2018107

Article Contents

2019, Volume 24, Issue 1: 257-272. Doi: 10.3934/dcdsb.2018107

This issue Previous Article A comparative study on nonlocal diffusion operators related to the fractional Laplacian Next Article A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

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
* Corresponding author: B. Reyes

Received: March 31, 2017

Revised: December 31, 2017

Early access: March 2018

Published: January 2019

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

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Abstract

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.

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

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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 $\gamma (t)$.

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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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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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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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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.