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Article: A Multiscale Reduced Basis Method for the Schrödinger Equation With Multiscale and Random Potentials

TitleA Multiscale Reduced Basis Method for the Schrödinger Equation With Multiscale and Random Potentials
Authors
KeywordsRandom Schrödinger equation
Multiscale reduced basis function
Optimization method
Quasi-Monte Carlo method
Anderson localization
Issue Date2020
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/mms/mms.htm
Citation
Multiscale Modeling & Simulation: a SIAM interdisciplinary journal, 2020, v. 18 n. 4, p. 1409-1434 How to Cite?
AbstractThe 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.
Persistent Identifierhttp://hdl.handle.net/10722/285077
ISSN
2023 Impact Factor: 1.9
2023 SCImago Journal Rankings: 1.028
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorChen, J-
dc.contributor.authorMa, D-
dc.contributor.authorZhang, Z-
dc.date.accessioned2020-08-07T09:06:25Z-
dc.date.available2020-08-07T09:06:25Z-
dc.date.issued2020-
dc.identifier.citationMultiscale Modeling & Simulation: a SIAM interdisciplinary journal, 2020, v. 18 n. 4, p. 1409-1434-
dc.identifier.issn1540-3459-
dc.identifier.urihttp://hdl.handle.net/10722/285077-
dc.description.abstractThe 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.-
dc.languageeng-
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/mms/mms.htm-
dc.relation.ispartofMultiscale Modeling & Simulation: a SIAM interdisciplinary journal-
dc.rights© 2020 Society for Industrial and Applied Mathematics. First Published in Multiscale Modeling & Simulation in v. 18 n. 4, published by the Society for Industrial and Applied Mathematics (SIAM).-
dc.subjectRandom Schrödinger equation-
dc.subjectMultiscale reduced basis function-
dc.subjectOptimization method-
dc.subjectQuasi-Monte Carlo method-
dc.subjectAnderson localization-
dc.titleA Multiscale Reduced Basis Method for the Schrödinger Equation With Multiscale and Random Potentials-
dc.typeArticle-
dc.identifier.emailZhang, Z: zhangzw@hku.hk-
dc.identifier.authorityZhang, Z=rp02087-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.1137/19M127389X-
dc.identifier.scopuseid_2-s2.0-85094879588-
dc.identifier.hkuros312372-
dc.identifier.hkuros316277-
dc.identifier.volume18-
dc.identifier.issue4-
dc.identifier.spage1409-
dc.identifier.epage1434-
dc.identifier.isiWOS:000600667900002-
dc.publisher.placeUnited States-
dc.identifier.issnl1540-3459-

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