{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:42:44Z","timestamp":1787330564594,"version":"build-2736575974"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-14-CE23-0005"],"award-info":[{"award-number":["ANR-14-CE23-0005"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>The solution of large sparse linear systems is one of the most time consuming kernels in many numerical simulations. The domain decomposition community has developed many efficient and robust methods in the last decades. While many of these solvers fall into the abstract Schwarz (aS) framework, their robustness was originally demonstrated on a case-by-case basis. In this paper, we propose a bound for the condition number of all deflated aS methods provided that the coarse grid consists of the assembly of local components that contain the kernel of some local operators. We show that classical results from the literature on particular instances of aS methods can be retrieved from this bound. We then show that such a coarse grid correction can be explicitly obtained algebraically via generalized eigenproblems, leading to a condition number independent of the number of domains. This result can be readily applied to retrieve or improve the bounds previously obtained via generalized eigenproblems in the particular cases of Neumann-Neumann (NN), additive Schwarz (AS), and optimized Robin, but it also generalizes them when applied with approximate local solvers. Interestingly, the proposed methodology turns out to be a comparison of the considered particular aS method with generalized versions of both NN and AS for tackling the lower and upper part of the spectrum, respectively. We furthermore show that the application of the considered grid corrections in an additive fashion is robust in the AS case although it is not robust for aS methods in general. In particular, the proposed framework allows for ensuring the robustness of the AS method applied on the Schur complement, either with deflation or additively, and with the freedom of relying on an approximate local Schur complement. Numerical experiments illustrate these statements.<\/jats:p>","DOI":"10.1137\/17m1153765","type":"journal-article","created":{"date-parts":[[2019,4,5]],"date-time":"2019-04-05T12:17:33Z","timestamp":1554466653000},"page":"417-439","source":"Crossref","is-referenced-by-count":9,"title":["Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers"],"prefix":"10.1137","volume":"40","author":[{"given":"Emmanuel","family":"Agullo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luc","family":"Giraud","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Louis","family":"Poirel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,4,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/17M1151158"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.crme.2010.11.005"},{"key":"atypb3","unstructured":"E. Agullo, L. Giraud, S. Nakov, and J. Roman,\n                      Hierarchical Hybrid Sparse Linear Solver for Multicore Platforms\n                      , Report, INRIA Bordeaux, 2016; also available online fromhttps:\/\/hal.inria.fr\/hal-01379227\/document."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899358194"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050446"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/nla.237"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/070706616"},{"key":"atypb8","unstructured":"Y.H. De Roeck and P. Le Tallec,\n                      Analysis and test of a local domain decomposition preconditioner\n                      , in Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Vol. 4, 1991."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502412887"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"V. Dolean, P. Jolivet, and F. Nataf,\n                      An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation\n                      , Vol. 144, SIAM, Philadelphia, 2015.","DOI":"10.1137\/1.9781611974065"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2011073"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1002\/nme.76"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620320604"},{"key":"atypb14","unstructured":"V. Frayss\u00e9 and L. Giraud,\n                      A Set of Conjugate Gradient Routines for Real and Complex Arithmetics\n                      , CERFACS technical report TR\/PA\/00\/47, 2000."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/090751190"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903425409"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0019-1"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/16M1060066"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8191(01)00141-7"},{"key":"atypb20","first-page":"1","author":"Jolivet P.","year":"2013","journal-title":"New York"},{"key":"atypb21","unstructured":"G. Karypis and V. Kumar,\n                      MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version4.0\n                      , 2009,http:\/\/www.cs.umn.edu\/ metis."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/15M1049610"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1553\/etna_vol49s1"},{"key":"atypb24","first-page":"75","volume":"45","author":"Klawonn A.","year":"2016","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb25","unstructured":"P. Le Tallec and M. Vidrascu,\n                      Generalized Neumann-Neumann preconditioners for iterative substructuring\n                      , in Proceedings of the Ninth International Symposium on Domain Decomposition Methods in Sciences and Engineering, 1998."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.1640090307"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/0727094"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"T. P. A. Mathew,\n                      Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations\n                      , Lect. Notes Comput. Sci. Eng. 61, Springer, New York, 2008.","DOI":"10.1007\/978-3-540-77209-5"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/100796376"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"A. Quarteroni and A. Valli,\n                      Domain Decomposition Methods for Partial Differential Equations\n                      , Oxford University Press, Oxford, 1999.","DOI":"10.1093\/oso\/9780198501787.001.0001"},{"key":"atypb31","first-page":"149","author":"Sarkis M.","year":"2003","journal-title":"New York"},{"key":"atypb32","unstructured":"N. Spillane,\n                      M\u00e9thodes de D\u00e9composition de Domaine Robustes Pour Les Probl\u00e8mes Sym\u00e9triques D\u00e9finis Positifs\n                      , Ph.D. thesis, Universit\u00e9 Pierre et Marie Curie, Paris, 2014."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-013-0576-y"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4534"},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"A. Toselli and O. Widlund,\n                      Domain Decomposition Methods-Algorithms and Theory\n                      , Springer Ser. Comput. Math. 34, Springer, New York, 2006.","DOI":"10.1007\/b137868"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1153765","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:12:27Z","timestamp":1787328747000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1153765"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1153765"],"URL":"https:\/\/doi.org\/10.1137\/17m1153765","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}