{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:14Z","timestamp":1787333114678,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We present a new \u201c$hp$\u201d parameter multidomain certified reduced basis (RB) method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (\u201ch\u201d-refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (\u201cp\u201d-refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the \u201ch\u201d-type and \u201cp\u201d-type stages of the new algorithm. We present illustrative numerical results for a convection-diffusion problem: the new \u201c$hp$\u201d-approach is considerably faster (respectively, more costly) than the standard \u201cp\u201d-type reduced basis method in the online (respectively, offline) stage.<\/jats:p>","DOI":"10.1137\/090780122","type":"journal-article","created":{"date-parts":[[2010,10,21]],"date-time":"2010-10-21T18:13:03Z","timestamp":1287684783000},"page":"3170-3200","source":"Crossref","is-referenced-by-count":121,"title":["An \"$hp$\" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations"],"prefix":"10.1137","volume":"32","author":[{"given":"Jens L.","family":"Eftang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anthony T.","family":"Patera","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Einar M.","family":"R\u00f8nquist","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,10,21]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.2514\/3.7539"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"D. Amsallem, J. Cortial, and C. Farhat,\n                      On-demand CFD-based aeroelastic predictions using a database of reduced-order bases and models\n                      , in 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, Orlando, FL, 2009, AIAA Paper 2009-800.","DOI":"10.2514\/6.2009-800"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.2514\/1.35374"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.08.006"},{"key":"R5","unstructured":"P. Binev, A. Cohen, W. Dahmen, R. DeVore, P. Guergana, and P. Wojtaszczyk,\n                      Convergence Rates for Greedy Algorithms in Reduced Basis Methods\n                      , in preparation."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/070688791"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2009.05.019"},{"key":"R8","unstructured":"A. Buffa, Y. Maday, A. T. Patera, C. Prud'homme, and G. Turinici,\n                      A priori convergence of the greedy algorithm for the parametrized reduced basis\n                      , M2AN Math. Model. Numer. Anal., submitted."},{"key":"R9","unstructured":"J. Eftang, B. Stamm, and A. Patera,\n                      Multi Parameter Element \u201chp\u201d Empirical Interpolation\n                      , in preparation."},{"key":"R10","unstructured":"J. L. Eftang, D. J. Knezevic, and A. T. Patera,\n                      An $hp$ certified reduced basis method for parametrized parabolic partial differential equations\n                      , Math. Comput. Model. Dyn. Syst., submitted."},{"key":"R11","unstructured":"J. L. Eftang, A. T. Patera, and E. M. R\u00f8nquist,\n                      An \u201c$hp$\u201d certified reduced basis method for parametrized parabolic partial differential equations\n                      , in Spectral and High Order Methods for Partial Differential Equations, J. S. Hesthaven, and E. M. R\u00f8nquist, eds., Lect. Notes Comput. Sci. Eng. 76, Springer, Berlin, to appear."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620070405"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2007031"},{"key":"R14","unstructured":"B. Haasdonk, M. Dihlmann, and M. Ohlberger,\n                      A Training Set and Multiple Bases Generation Approach for Parametrized Model Reduction Based on Adaptive Grids in Parameter Space\n                      , Technical report 28, SRC SimTech, available online from http:\/\/www.simtech.uni-stuttgart.de\/publikationen\/(preprints) (2010); Math. Comput. Model. Dyn. Syst., submitted."},{"key":"R15","unstructured":"B. Haasdonk and M. Ohlberger,\n                      Adaptive basis enrichment for the reduced basis method applied to finite volume schemes\n                      , in Finite Volumes for Complex Applications V, ISTE, London, 2008, pp. 471\u2013478."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2008001"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2007.09.019"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/090759239"},{"key":"R19","unstructured":"N. C. Nguyen, G. Rozza, D. B. P. Huynh, and A. T. Patera,\n                      Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs: Application to real-time Bayesian parameter estimation\n                      , in Computational Methods for Large Scale Inverse Problems and Uncertainty Quantifications, Biegler, Biro, Ghattas, Heinkenschloss, Keyes, Mallick, Tenorio, van Bloemen Waanders, and Willcox, eds., John Wiley & Sons, Chichester, UK, to appear."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-009-0005-x"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.2514\/3.50778"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1985-0804937-0"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(93)90050-3"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/s11831-008-9019-9"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1080\/10407790802424204"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"K. Veroy, C. Prud'homme, D. V. Rovas, and A. T. Patera,\n                      A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations\n                      , in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, 2003, AIAA Paper 2003-3847.","DOI":"10.2514\/6.2003-3847"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090780122","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:33:26Z","timestamp":1787330006000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090780122"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":26,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090780122"],"URL":"https:\/\/doi.org\/10.1137\/090780122","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}