@article {26621,
	title = {   A branching random walk method for many-body Wigner quantum dynamics},
	journal = {Numerical Mathematics: Theory, Methods and Applications},
	volume = {12},
	year = {2019},
	pages = {21-71},
	abstract = {A branching random walk algorithm for many-body Wigner\&nbsp;equations and its numerical applications for quantum dynamics in phase space\&nbsp;are proposed and analyzed in this paper.\&nbsp;Using an auxiliary function, the truncated Wigner\&nbsp;equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations.\&nbsp;We prove that the first moment of a branching random walk is the solution for the adjoint equation.\&nbsp;With the help of the additional degree of freedom offered by the auxiliary function,\&nbsp;we are able to produce a weighted-particle implementation of the branching random walk.\&nbsp;In contrast to existing signed-particle implementations,\&nbsp;this weighted-particle one shows a key capacity of variance reduction\&nbsp;by increasing the constant auxiliary function and has no time discretization errors.\&nbsp;Several canonical numerical experiments on the 2D Gaussian barrier scattering\&nbsp;and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency,\&nbsp;and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.},
	url = {https://www.global-sci.org/uploads/Issue/NMTMA/v12n1/121_21.pdf},
	author = {Sihong Shao and Yunfeng Xiong}
}
