Volume 25, Issue 3
The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation

Yunfeng Xiong & Sihong Shao

Commun. Comput. Phys., 25 (2019), pp. 871-910.

Published online: 2018-11

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  • Abstract

To implement the Wigner branching random walk, the particle carrying a signed weight, either −1 or +1, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from −1 to +1. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adjust the skewness of particle weights. Performance evaluations on the Gaussian barrier scattering (2D) and a Helium-like system (4D) demonstrate the feasibility of both strategies and the variance reduction property of the second approach. We provide an improvement of the first signed-particle implementation that partially alleviates the restriction on the time step and perform a thorough theoretical and numerical comparison among all the existing signed-particle implementations. Details on implementing the importance sampling according to the quasi-probability density and an efficient resampling or particle reduction are also provided.

  • Keywords

Wigner equation, branching random walk, signed particle, bootstrapping, weighted particle, Monte Carlo method, quantum dynamics, importance sampling, resampling, particle reduction.

  • AMS Subject Headings

60J85, 81S30, 45K05, 65M75, 82C10, 81V70, 81Q05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-871, author = {}, title = {The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {3}, pages = {871--910}, abstract = {

To implement the Wigner branching random walk, the particle carrying a signed weight, either −1 or +1, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from −1 to +1. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adjust the skewness of particle weights. Performance evaluations on the Gaussian barrier scattering (2D) and a Helium-like system (4D) demonstrate the feasibility of both strategies and the variance reduction property of the second approach. We provide an improvement of the first signed-particle implementation that partially alleviates the restriction on the time step and perform a thorough theoretical and numerical comparison among all the existing signed-particle implementations. Details on implementing the importance sampling according to the quasi-probability density and an efficient resampling or particle reduction are also provided.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0141}, url = {http://global-sci.org/intro/article_detail/cicp/12832.html} }
TY - JOUR T1 - The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation JO - Communications in Computational Physics VL - 3 SP - 871 EP - 910 PY - 2018 DA - 2018/11 SN - 25 DO - http://dor.org/10.4208/cicp.OA-2018-0141 UR - https://global-sci.org/intro/article_detail/cicp/12832.html KW - Wigner equation, branching random walk, signed particle, bootstrapping, weighted particle, Monte Carlo method, quantum dynamics, importance sampling, resampling, particle reduction. AB -

To implement the Wigner branching random walk, the particle carrying a signed weight, either −1 or +1, is more friendly to data storage and arithmetic manipulations than that taking a real-valued weight continuously from −1 to +1. The former is called a signed particle and the latter a weighted particle. In this paper, we propose two efficient strategies to realize the signed-particle implementation. One is to interpret the multiplicative functional as the probability to generate pairs of particles instead of the incremental weight, and the other is to utilize a bootstrap filter to adjust the skewness of particle weights. Performance evaluations on the Gaussian barrier scattering (2D) and a Helium-like system (4D) demonstrate the feasibility of both strategies and the variance reduction property of the second approach. We provide an improvement of the first signed-particle implementation that partially alleviates the restriction on the time step and perform a thorough theoretical and numerical comparison among all the existing signed-particle implementations. Details on implementing the importance sampling according to the quasi-probability density and an efficient resampling or particle reduction are also provided.

Yunfeng Xiong & Sihong Shao. (2020). The Wigner Branching Random Walk: Efficient Implementation and Performance Evaluation. Communications in Computational Physics. 25 (3). 871-910. doi:10.4208/cicp.OA-2018-0141
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