{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:33:12Z","timestamp":1787329992891,"version":"build-2736575974"},"reference-count":81,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["411724963"],"award-info":[{"award-number":["411724963"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2024,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\u00f6dinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in nonuniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain\u2019s anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.<\/jats:p>\n                  <jats:p>Reproducibility of computational results. This paper has been awarded the \u201cSIAM Reproducibility Badge: Code and data available\u201d as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https:\/\/doi.org\/10.5281\/zenodo.11072863 . [Formula: see text]<\/jats:p>","DOI":"10.1137\/23m161848x","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T04:10:07Z","timestamp":1727755807000},"page":"A3067-A3093","source":"Crossref","is-referenced-by-count":0,"title":["A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\u00f6dinger Eigenstates in Anisotropically Expanding Domains"],"prefix":"10.1137","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5460-5425","authenticated-orcid":true,"given":"Lambert","family":"Theisen","sequence":"first","affiliation":[{"name":"Corresponding author.\u00a0Applied and Computational Mathematics, RWTH Aachen University, 52062 Aachen, Germany, and Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569, Stuttgart, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3375-483X","authenticated-orcid":true,"given":"Benjamin","family":"Stamm","sequence":"additional","affiliation":[{"name":"Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569 Stuttgart, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,10,1]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/21M1446320"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/22M1469833"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(99)00112-7"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-021-01216-5"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2022036"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-024-02612-3"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.21105\/joss.02520"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719581"},{"key":"ref9","volume-title":"Parallel Computing: Architectures, Algorithms and Applications","author":"Barrault M.","year":"2008"},{"key":"ref10","author":"Bastian P.","year":"2022","journal-title":"SIAM J. Sci. Comput."},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-8758-5_8"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/070701728"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502400650"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/141000671"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drac036"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759732678X"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-22340-2"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.107.155403"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3897"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/20M1332864"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1063\/1.4816767"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1038\/nature26160"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3066"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.parco.2007.12.001"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1137\/21M1466748"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1137\/16M1065215"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1137\/17M1115885"},{"key":"ref29","volume":"49","author":"Ciaramella G.","year":"2018","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611976908"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-56750-7_16"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.5802\/smai-jcm.61"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1137\/15M102887X"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974065"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.2478\/cmam-2012-0027"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1137\/23M1560938"},{"key":"ref37","unstructured":"M. Dryja and O. Widlund, An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Technical report, Courant Institute, New York, 1987."},{"key":"ref38","unstructured":"M. A. Freitag, Inner-Outer Iterative Methods for Eigenvalue Problems - Convergence and Preconditioning, Ph.D. thesis, University of Bath, 2007."},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-006-0100-1"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2021.108007"},{"key":"ref41","volume":"8","author":"Galvis J.","year":"2010","journal-title":"Multiscale Model. Simul."},{"key":"ref42","volume":"8","author":"Galvis J.","year":"2010","journal-title":"Multiscale Model. Simul."},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-93873-8_28"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1038\/nphys1836"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61798-0"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.56021\/9781421407944"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1553\/etna_vol60s169"},{"key":"ref48","unstructured":"M. Hassan. Mathematical Analysis of Boundary Integral Equations and Domain Decomposition Methods with Applications in Polarisable Electrostatics, Ph.D. thesis, RWTH Aachen University, 2020."},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110165"},{"key":"ref50","volume":"48","author":"Heinlein A.","year":"2018","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1137\/18M1220613"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202523500343"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1137\/130921520"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1137\/22M1516300"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/18M1230463"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111127"},{"key":"ref57","volume":"3","author":"Herbst M. F.","year":"2021","journal-title":"Proc. JuliaCon Conferences"},{"key":"ref58","volume":"49","author":"Hestenes M.","year":"1952","journal-title":"J. Res. Natl. Inst. Stand. Technol."},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2154"},{"key":"ref60","unstructured":"G. Karypis and V. Kumar, METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, Technical report, University of Minnesota, 1997."},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970944"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1137\/140997907"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-6332-2_11"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-33287-6"},{"key":"ref65","unstructured":"S. H. Lui, Some recent results on domain decomposition methods for eigenvalue problems, in Proceedings of the Ninth International Conference on Domain Decomposition Methods, 1996."},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(99)00326-X"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-77209-5"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1038\/s41565-017-0035-5"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1137\/0724027"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2024.113097"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1137\/20M1342872"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.23.69"},{"key":"ref73","unstructured":"N. Spillane. Robust Domain Decomposition Methods for Symmetric Positive Definite Problems, Ph.D. thesis, Universit\u00e9 Pierre et Marie Curie - Paris VI, 2014."},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-013-0576-y"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1137\/21M1456005"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1137\/15M1049956"},{"key":"ref77","author":"Theisen L.","year":"2024","journal-title":"ddEigenLab.jl: Domain-Decomposition Eigenvalue Problem Lab (v0.4)"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1007\/b137868"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.109.160401"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2020.06.014"},{"key":"ref81","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.90.187401"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M161848X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:43:10Z","timestamp":1787326990000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M161848X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,1]]},"references-count":81,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10,31]]}},"alternative-id":["10.1137\/23M161848X"],"URL":"https:\/\/doi.org\/10.1137\/23m161848x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,1]]}}}