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Volume 14, Issue 2
Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime

Xiaoyu Lei & Zhennan Zhou

DOI: 10.4208/nmtma.OA-2020-0137

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 321-354.

Published online: 2021-01

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  • Abstract

With the path integral approach, the thermal average in multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.

  • Keywords

Multi-level Monte Carlo method, Langevin sampling, path integral molecular dynamics, thermal average.

  • AMS Subject Headings

65C05, 82C31, 82B10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-321, author = {Lei , Xiaoyu and Zhou , Zhennan}, title = {Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {321--354}, abstract = {

With the path integral approach, the thermal average in multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0137}, url = {http://global-sci.org/intro/article_detail/nmtma/18602.html} }
TY - JOUR T1 - Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime AU - Lei , Xiaoyu AU - Zhou , Zhennan JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 321 EP - 354 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0137 UR - https://global-sci.org/intro/article_detail/nmtma/18602.html KW - Multi-level Monte Carlo method, Langevin sampling, path integral molecular dynamics, thermal average. AB -

With the path integral approach, the thermal average in multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.

Lei , Xiaoyu and Zhou , Zhennan. (2021). Multi-Level Monte Carlo Path Integral Molecular Dynamics for Thermal Average Calculation in the Nonadiabatic Regime. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 321-354. doi:10.4208/nmtma.OA-2020-0137
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