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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:

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##### References:
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 $\gamma (t)$.
Two realizations of moving-mesh configurations.
Simulated density profiles at three discrete times steps
Total CPU time to simulate the expected value of the QoI using the MC and MLMC algorithms with $\epsilon = [N_{MC}]^{-1/2}$.
Variance for four levels of $Q^{(\ell)}$ and $Q^{(\ell)} - Q^{(\ell-1)}$
Adaptively chosen values of $N_{{\rm{MLMC}}}^{(\ell)},~\ell = 0,1, 2, 3$ for four distinct choices of $\epsilon$ as in Table 3.
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
$\mathbb{E}_{\rm{mc}} \left[Q; N_{MC} \right]$ 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
$\mathbb{E}_{{\rm{MLMC}}}^{(3)}\left[Q^{(3)}\right]$ 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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