{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T04:29:46Z","timestamp":1772166586969,"version":"3.50.1"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T00:00:00Z","timestamp":1604016000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T00:00:00Z","timestamp":1604016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000785","name":"Education, Audiovisual and Culture Executive Agency","doi-asserted-by":"publisher","award":["FPA 2013-0043"],"award-info":[{"award-number":["FPA 2013-0043"]}],"id":[{"id":"10.13039\/501100000785","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["DPI2017-85139-C2-2-R"],"award-info":[{"award-number":["DPI2017-85139-C2-2-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002809","name":"Generalitat de Catalunya","doi-asserted-by":"publisher","award":["2017-SGR-1278"],"award-info":[{"award-number":["2017-SGR-1278"]}],"id":[{"id":"10.13039\/501100002809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010661","name":"Horizon 2020 Framework Programme","doi-asserted-by":"publisher","award":["H2020-RISE MATH-ROCKS GA no 777778"],"award-info":[{"award-number":["H2020-RISE MATH-ROCKS GA no 777778"]}],"id":[{"id":"10.13039\/100010661","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv. Model. and Simul. in Eng. Sci."],"published-print":{"date-parts":[[2020,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Reduced order methods are powerful tools for the design and analysis of sophisticated systems, reducing computational costs and speeding up the development process. Among these reduced order methods, the Proper Generalized Decomposition is a well-established one, commonly used to deal with multi-dimensional problems that often suffer from the\n                    <jats:italic>curse of dimensionality<\/jats:italic>\n                    . Although the PGD method has been around for some time now, it still lacks mechanisms to assess the quality of the solutions obtained. This paper explores the dual error analysis in the scope of the PGD, using complementary solutions to compute error bounds and drive an adaptivity process, applied to a simple 1D problem. The energy of the error obtained from the dual analysis is used to determine the quality of the PGD approximations. We define a new adaptivity indicator based on the energy of the error and use it to drive parametric h- and p- adaptivity processes. The results are positive, with the indicator accurately capturing the parameter that will lead to lowest errors.\n                  <\/jats:p>","DOI":"10.1186\/s40323-020-00180-3","type":"journal-article","created":{"date-parts":[[2020,10,30]],"date-time":"2020-10-30T07:03:45Z","timestamp":1604041425000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Error estimation and adaptivity for PGD based on complementary solutions applied to a simple 1D problem"],"prefix":"10.1186","volume":"7","author":[{"given":"Jonatha","family":"Reis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 Paulo","family":"Moitinho de Almeida","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6464-6407","authenticated-orcid":false,"given":"Pedro","family":"D\u00edez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergio","family":"Zlotnik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,10,30]]},"reference":[{"key":"180_CR1","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s12289-009-0647-x","volume":"3","author":"A Ammar","year":"2010","unstructured":"Ammar A. The proper generalized decomposition: a powerful tool for model reduction. Int J Mater Form. 2010;3:89\u2013102.","journal-title":"Int J Mater Form"},{"key":"180_CR2","doi-asserted-by":"publisher","first-page":"2032","DOI":"10.1016\/j.cma.2011.02.019","volume":"200","author":"P Ladev\u00e8ze","year":"2011","unstructured":"Ladev\u00e8ze P, Chamoin L. On the verification of model reduction methods based on the proper generalized decomposition. Comput Methods Appl Mech Eng. 2011;200:2032\u201347.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"180_CR3","doi-asserted-by":"publisher","first-page":"1872","DOI":"10.1016\/j.cma.2010.02.012","volume":"199","author":"A Ammar","year":"2010","unstructured":"Ammar A, Chinesta F, Diez P, Huerta A. An error estimator for separated representations of highly multidimensional models. Comput Methods Appl Mech Eng. 2010;199:1872\u201380.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"180_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-02865-1","volume-title":"The proper generalized decomposition for advanced numerical simulations: a primer","author":"F Chinesta","year":"2014","unstructured":"Chinesta F, Keunings R, Leygue A. The proper generalized decomposition for advanced numerical simulations: a primer. Cham: Springer; 2014."},{"key":"180_CR5","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1186\/s40323-015-0052-6","volume":"2","author":"S Zlotnik","year":"2015","unstructured":"Zlotnik S, D\u00edez P, Gonzalez D, Cueto E, Huerta A. Effect of the separated approximation of input data in the accuracy of the resulting PGD solution. Advanced modeling and simulation in engineering sciences. 2015;2:28.","journal-title":"Advanced modeling and simulation in engineering sciences"},{"key":"180_CR6","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/j.cma.2013.09.003","volume":"268","author":"A Ammar","year":"2014","unstructured":"Ammar A, Huerta A, Chinesta F, Cueto E, Leygue A. Parametric solutions involving geometry: a step towards efficient shape optimization. Comput Methods Appl Mech Eng. 2014;268:178\u201393.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"180_CR7","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1007\/s00466-015-1246-y","volume":"57","author":"A Courard","year":"2016","unstructured":"Courard A, N\u00e9ron D, Ladev\u00e8ze P, Ballere L. Integration of PGD-virtual charts into an engineering design process. Comput Mech. 2016;57:637\u201351.","journal-title":"Comput Mech"},{"key":"180_CR8","doi-asserted-by":"publisher","first-page":"961","DOI":"10.1002\/nme.4490","volume":"94","author":"JPM de Almeida","year":"2013","unstructured":"de Almeida JPM. A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics. Int J Numer Methods Eng. 2013;94:961\u201384.","journal-title":"Int J Numer Methods Eng"},{"key":"180_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/9781118925782","volume-title":"Equilibrium finite element formulations","author":"JPM Almeida","year":"2017","unstructured":"Almeida JPM, Maunder EAW. Equilibrium finite element formulations. Chichester: Wiley; 2017. p. 1\u2013274."},{"key":"180_CR10","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1090\/qam\/25902","volume":"5","author":"W Prager","year":"1947","unstructured":"Prager W, Synge JL. Approximations in elasticity based on the concept of function space. Q Appl Math. 1947;5:241\u201369.","journal-title":"Q Appl Math"},{"key":"180_CR11","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.cma.2015.03.026","volume":"295","author":"D Modesto","year":"2015","unstructured":"Modesto D, Zlotnik S, Huerta A. Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation. Comput Methods Appl Mech Eng. 2015;295:127\u201349.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"180_CR12","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.cma.2017.08.047","volume":"327","author":"L Chamoin","year":"2017","unstructured":"Chamoin L, Pled F, Allier P-E, Ladev\u00e8ze P. A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems. Comput Methods Appl Mech Eng. 2017;327:118\u201346.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"180_CR13","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1002\/nme.6452","volume":"10","author":"J Reis","year":"2019","unstructured":"Reis J, Almeida JP, D\u00edez P, Zlotnik S. Error estimation for proper generalized decomposition solutions: a dual approach. Int J Numer Methods Eng. 2019;10:12. https:\/\/doi.org\/10.1002\/nme.6452 .","journal-title":"Int J Numer Methods Eng"},{"key":"180_CR14","doi-asserted-by":"crossref","unstructured":"Reis J, Almeida JPM, D\u00edez P, Zlotnick S. Error estimation for pgd solutions: Dual analysis and adaptivity for quantities of interest (2020). Submitted for publication","DOI":"10.1002\/nme.6559"}],"container-title":["Advanced Modeling and Simulation in Engineering Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-020-00180-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1186\/s40323-020-00180-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-020-00180-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T15:11:09Z","timestamp":1669389069000},"score":1,"resource":{"primary":{"URL":"https:\/\/amses-journal.springeropen.com\/articles\/10.1186\/s40323-020-00180-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,30]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["180"],"URL":"https:\/\/doi.org\/10.1186\/s40323-020-00180-3","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-29637\/v1","asserted-by":"object"}]},"ISSN":["2213-7467"],"issn-type":[{"value":"2213-7467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,30]]},"assertion":[{"value":"14 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"42"}}