六维相空间的Wigner-Boltzmann方程的高效随机粒子算法

Wigner-Boltzmann方程是描述半导体中电子的量子力学行为的基本数学模型，然而受维数灾难的限制，现有的数值方法很难实现六维的数值模拟。本项目旨在开发高效的随机粒子方法，解决六维Wigner-Boltzmann方程数值模拟的高维、高度非线性和非局部的困难，并实现三纳米制程半导体材料的数值模拟。本项目拟从非线性偏微分方程的随机解释、随机模型的方差缩减、高维粒子重采样技术三个角度来进行研究。首先结合非线性马尔可夫半群理论和指数积分子理论，对一般的拟微分算子采取正负分裂的方式，建立方程与带权粒子系统的分枝随机游走之间的数学联系。然后，鉴于拟微分算子的振荡结构会带来正负权重抵消，造成方差指数增长的问题，本项目利用振荡积分算子的低频和高频部分局域化的性质，控制随机模型的方差。最后，借鉴高维密度估计理论和偏差理论的最新进展，构建高维空间的粒子重采样技术，以此控制粒子数随时间指数增长的问题。

The Wigner-Boltzmann equation is a basic mathematical model to describe the quantum mechanical behaviors of electrons in semiconductor devices. However, its numerical simulations in six-dimensional phase space are still hampered by the curse of dimensionality. This project aims at developing an efficient stochastic particle method for the six-dimensional Wigner-Boltzmann equation and overcoming the difficulties induced by high dimensionality, strong nonlinearity and nonlocality, as well as discussing its applications in simulating the semiconductor devices at 3nm technology nodes. To this end, we need to study the stochastic interpretation of nonlinear partial differential equation, variance reduction in stochastic models and high-dimensional particle resampling technique. First, we need to establish the rigorous mathematical connection between PDE and the stochastic branching random walk of weighted interacting particle system. Our approach exploits the combination of the nonlinear Markov semigroup theory, the exponential integrators and the splitting technique for general pseudo-differential operators. Second, regarding the fact that the oscillatory nature of the pseudo-differential operator may lead to an exponential growth of stochastic variances due to the near-cancelation of positive and negative weights, we utilize the localization property of the low-frequency and high-frequency parts of the pseudo-differential operator to control the variances. Finally, we establish the particle resampling technique in high-dimensional space to control the exponential growth of particle number in long-time simulation.