{"title":"A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential","authors":"T. Jahnke, B. Stein","doi":"10.1137/21m1440517","DOIUrl":null,"url":null,"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","PeriodicalId":56064,"journal":{"name":"Siam-Asa Journal on Uncertainty Quantification","volume":"27 1","pages":"1753-1780"},"PeriodicalIF":2.1000,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siam-Asa Journal on Uncertainty Quantification","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1137/21m1440517","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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
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.
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