{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:35:25Z","timestamp":1787337325990,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/J500197\/1"],"award-info":[{"award-number":["EP\/J500197\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Proper generalized decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs, which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric, and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGDs provide a much stabler algorithm than an equivalent Galerkin PGDs, and we provide proofs of convergence of the algorithms.<\/jats:p>","DOI":"10.1137\/15m1049269","type":"journal-article","created":{"date-parts":[[2017,8,1]],"date-time":"2017-08-01T11:49:21Z","timestamp":1501588161000},"page":"A1366-A1388","source":"Crossref","is-referenced-by-count":4,"title":["Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems"],"prefix":"10.1137","volume":"39","author":[{"given":"Thomas L. D.","family":"Croft","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Timothy N.","family":"Phillips","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717495"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160120405"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160170104"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnnfm.2006.07.007"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1985-0771030-5"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165003"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.112329799"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2014019"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1257573-4"},{"key":"atypb10","volume-title":"Least-Squares Finite Element Methods","author":"Bochev P. B.","year":"2009"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000182"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/0731091"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299527299X"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202511005799"},{"key":"atypb16","first-page":"95","volume-title":"ESAIM Proc. 41","author":"Canc\u00e8s E.","year":"2013"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539794273368"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnnfm.2010.12.012"},{"key":"atypb19","unstructured":"T. L. D. Croft,\n                      Proper Generalised Decompositions: Theory and Applications\n                      , Ph.D. thesis, School of Mathematics, Cardiff University, Cardiff, Wales, 2015."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595294526"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02288367"},{"key":"atypb22","volume-title":"Finite Elements and Fast Iterative Solvers","author":"Elman H.","year":"2005"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.12.003"},{"key":"atypb24","volume-title":"Convex Analysis and Minimisation Algorithms I","author":"Hiriart-Urruty J.-B.","year":"1993"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479800368354"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-009-9071-1"},{"key":"atypb27","unstructured":"Y. Maday, A. T. Patera, and E. M. R\u00f8nquist,\n                      The $P_N-P_{N-2}$ Method for the Approximation of the Stokes Problem\n                      , Technical Report 92025, Laboratoire d'Analyse Num\u00e9rique, Universit\u00e9 Pierre et Marie Curie, Paris, 1992."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.01.009"},{"key":"atypb29","unstructured":"M. M. J. Proot,\n                      The Least-Squares Spectral Element Method\n                      , Ph.D. thesis, Department of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands, 2003."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1142\/p499"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018900431309"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1049269","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:58:01Z","timestamp":1787335081000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1049269"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/15M1049269"],"URL":"https:\/\/doi.org\/10.1137\/15m1049269","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}