{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:44Z","timestamp":1787337224349,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the Galerkin matrix. We compare the popular mean-based preconditioner with the proposed preconditioner which\u2014in contrast to the mean-based construction\u2014makes use of the entire information contained in the Galerkin matrix. We report on results of test problems, where the random diffusion coefficient is given in terms of a truncated Karhunen\u2013Lo\u00e8ve expansion or is a lognormal random field.<\/jats:p>","DOI":"10.1137\/080742853","type":"journal-article","created":{"date-parts":[[2010,3,24]],"date-time":"2010-03-24T18:10:36Z","timestamp":1269454236000},"page":"923-946","source":"Crossref","is-referenced-by-count":66,"title":["A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations"],"prefix":"10.1137","volume":"32","author":[{"given":"Elisabeth","family":"Ullmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,3,24]]},"reference":[{"key":"R1","unstructured":"R. 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A. Zang,\n                      Spectral Methods. Fundamentals in Single Domains\n                      , Springer-Verlag, Berlin, 2006.","DOI":"10.1007\/978-3-540-30726-6"},{"key":"R8","unstructured":"R. Courant and D. Hilbert,\n                      Methods of Mathematical Physics\n                      , Vol. I, Interscience, New York, 1953."},{"key":"R9","unstructured":"M. Deb,\n                      Solution of Stochastic Partial Differential Equations (SPDEs) Using Galerkin Method: Theory and Applications\n                      , Ph.D. thesis, The University of Texas, Austin, TX, 2000."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/070705817"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.04.008"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"R. Ghanem and P. 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Keese,\n                      Numerical Solution of Systems with Stochastic Uncertainties\u2014A General Purpose Framework for Stochastic Finite Elements\n                      , Ph.D. thesis, Fachbereich f\u00fcr Mathematik und Informatik, Technische Universit\u00e4t Braunschweig, Braunschweig, Germany, 2004."},{"key":"R19","unstructured":"M. Lo\u00e8ve,\n                      Probability Theory\n                      , vol. II, 4th ed., Springer-Verlag, New York, Heidelberg, Berlin, 1978."},{"key":"R20","doi-asserted-by":"crossref","unstructured":"P. Malliavin,\n                      Stochastic Analysis\n                      , Springer-Verlag, Berlin, 1997.","DOI":"10.1007\/978-3-642-15074-6"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.05.027"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/050631707"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02547521"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"B. \u00d8ksendal,\n                      Stochastic Differential Equations\n                      , 5th ed., Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-662-03620-4"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/S0965-9978(00)00034-X"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"C. E. Powell and H. C. Elman,\n                      Block-diagonal preconditioning for spectral stochastic finite-element systems\n                      , IMA J. Numer. 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R. de Moor, eds., Kluwer, Dordrecht, 1993, pp. 293\u2013314.","DOI":"10.1007\/978-94-015-8196-7_17"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501387826"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080742853","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:26Z","timestamp":1787334626000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080742853"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080742853"],"URL":"https:\/\/doi.org\/10.1137\/080742853","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}