A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

IF 2.1 3区工程技术 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Siam-Asa Journal on Uncertainty Quantification Pub Date : 2022-12-20 DOI:10.1137/21m1440517

T. Jahnke, B. Stein

引用次数: 0

Abstract

We propose and analyze a numerical method for time-dependent linear Schrödinger equations with 5 uncertain parameters in both the potential and the initial data. The random parameters are dis6 cretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are ap7 proximated with the Strang splitting method. The computational work is reduced by a multi-level 8 strategy, i.e. by combining information obtained from sample solutions computed on different re9 finement levels of the discretization. We prove new error bounds for the time discretization which 10 take the finite regularity in the stochastic variable into account, and which are crucial to obtain 11 convergence of the multi-level approach. The predicted cost savings of the multi-level stochastic 12 collocation method are verified by numerical examples. 13

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具有随机势的Schrödinger方程的多层随机配置方法

我们提出并分析了具有5个不确定参数的时变线性Schrödinger方程的数值方法。在稀疏网格上采用随机配置的方法对随机参数进行离散，并采用Strang分裂法对节点上的样本解进行近似。通过多级策略减少了计算工作量，即通过组合在不同的离散化精细水平上计算的样本解获得的信息。我们证明了考虑随机变量的有限正则性的时间离散化的新的误差界，这对于得到多阶方法的收敛性是至关重要的。通过数值算例验证了多级随机12配置法所预测的成本节约。13

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Siam-Asa Journal on Uncertainty Quantification Mathematics-Statistics and Probability

CiteScore

3.70

自引率

0.00%

发文量

期刊介绍： SIAM/ASA Journal on Uncertainty Quantification (JUQ) publishes research articles presenting significant mathematical, statistical, algorithmic, and application advances in uncertainty quantification, defined as the interface of complex modeling of processes and data, especially characterizations of the uncertainties inherent in the use of such models. The journal also focuses on related fields such as sensitivity analysis, model validation, model calibration, data assimilation, and code verification. The journal also solicits papers describing new ideas that could lead to significant progress in methodology for uncertainty quantification as well as review articles on particular aspects. The journal is dedicated to nurturing synergistic interactions between the mathematical, statistical, computational, and applications communities involved in uncertainty quantification and related areas. JUQ is jointly offered by SIAM and the American Statistical Association.