Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1409-1434, January 2020.
The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wave function develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. To address this problem, in this paper we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial grid size is only proportional to the semiclassical parameter and (under suitable conditions) an almost first-order convergence rate is achieved in the random space with respect to the sample number. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for the Schrödinger equation with correlated random potentials in both 1-dimensional and 2-dimensional space.

中文翻译：

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具有多尺度和随机势的Schrödinger方程的多尺度降阶基础方法

2020年1月，《多尺度建模与仿真》，第18卷，第4期，第1409-1434页。
研究异质量子系统中的电子动力学时，经常会出现具有多尺度和随机势的半经典Schrödinger方程。随着时间的流逝，波函数在物理空间和随机空间中都产生了高频振荡，这对数值方法提出了严峻的挑战。为了解决这个问题，本文提出了一种多尺度约简方法，在物理空间中使用优化方法和适当的正交分解方法构造多尺度约简函数，并在随机空间中采用准蒙特卡罗方法。我们的方法经过验证是有效的：空间网格的大小仅与半经典参数成正比，并且（在合适的条件下）相对于样本数，在随机空间中获得了几乎一阶的收敛速度。通过数值证明，研究了该方法的几个理论方面，包括如何确定多尺度缩减基的构造中的样本数量以及收敛性分析。此外，我们研究了一维和二维空间中具有相关随机势的薛定ding方程的安德森局部化现象。