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Original Articles

An hp certified reduced basis method for parametrized parabolic partial differential equations

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Pages 395-422 | Received 05 Apr 2010, Accepted 07 Sep 2010, Published online: 28 Jul 2011

References

  • Noor , A.K. and Peters , J.M. 1980 . Reduced basis technique for nonlinear analysis of structures . AIAA J. , 18 : 455 – 462 .
  • Almroth , B.O. , Stern , P. and Brogan , F.A. 1978 . Automatic choice of global shape functions in structural analysis . AIAA J. , 16 : 525 – 528 .
  • Haasdonk , B. and Ohlberger , M. 2008 . Reduced basis method for finite volume approximations of parametrized linear evolution equations . Math. Model. Numer. Anal. , 42 : 277 – 302 .
  • Grepl , M.A. and Patera , A.T. 2005 . A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations . M2AN Math. Model. Numer. Anal. , 39 : 157 – 181 .
  • Rozza , G. , Huynh , D.B.P. and Patera , A.T. 2008 . Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations . Arch. Comput. Methods Eng. , 15 : 229 – 275 .
  • Nguyen , N.C. , Rozza , G. , Huynh , D.B.P. and Patera , A.T. 2010 . “ Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs: Application to real-time Bayesian Parameter estimation ” . In Large-Scale Inverse Problems and Quantification of Uncertainty, Wiley Series in Computational Statistics , Edited by: Biegler , L. , Biro , G. , Ghattas , O. , Heinkenschloss , M. , Keyes , D. , Mallick , B. , Tenorio , L. , van Bloemen Waanders , B. and Willcox , K. John Wiley & Sons, Chichester .
  • Eftang , J.L. , Patera , A.T. and Ronquist , E.M. 2010 . An ‘hp’ certified reduced basis method for parametrized elliptic partial differential equations . SIAM J. Sci. Comput. , 32 : 3170 – 3200 .
  • Haasdonk , B. , Dihlmann , M. and Ohlberger , M. 2010 . A training set and multiple bases generation approach for parametrized model reduction based on adaptive grids in parameter space . Tech. Rep. 28, SRC SimTech ,
  • Amsallem , D. , Cortial , J. and Farhat , C. January 2009 . “ On-demand CFD-based aeroelastic predictions using a database of reduced-order bases and models ” . In 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition January , Orlando, FL
  • Eftang , J.L. , Patera , A.T. and ønquist , E.M.R . 2009 . “ An hp certified reduced basis method for parametrized parabolic partial differential equations ” . In Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering , Edited by: Hesthaven , J.S. and ønquist , E.M.R . Vol. 76 , 179 – 187 . Trondheim, , Norway : Springer .
  • Knezevi , D.J. , Nguyen , N.C. and Patera , A.T. 2010 . “ Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations ” . In M3AS, Accepted c
  • Nguyen , N.C. , Rozza , G. and Patera , A.T. 2009 . Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation . Calcolo , 46 : 157 – 185 .
  • Barrault , M. , Maday , Y. , Nguyen , N.C. and Patera , A.T. 2004 . An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations . C. R. Math. Acad. Sci. Paris , 339 : 667 – 672 .
  • Eftang , J.L. , Grepl , M.A. and Patera , A.T. 2010 . A posteriori error bounds for the empirical interpolation method . C. R. Math. , 348 : 575 – 579 .
  • Huynh , D.B.P. , Rozza , G. , Sen , S. and Patera , A.T. 2007 . A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants . C. R. Math. Acad. Sci. Paris , 345 : 473 – 478 .
  • Sirovich , L. 1987 . Turbulence and the dynamics of coherent structures. Part I. Coherent structures . Quart. Appl. Math. , 45 : 561 – 571 .
  • Knezevic , D.J. and Patera , A.T. 2010 . “ A certified reduced basis method for the fokker-planck equation of dilute polymeric fluids: fene dumbbells in extensional flow ” . In SIAM J. Sci. Comput 793 – 817 .
  • Quarteroni , A. and Valli , A. 1994 . “ Numerical approximation of partial differential equations ” . In Springer Series in Computational Mathematics , Vol. 23 , Berlin : Springer-Verlag .
  • Knezevic , D.J. and Peterson , J.W. 2010 . “ A high-performance parallel implementation of the certified reduced basis method ” . In Comput. Methods Appl. Mech. Eng., Submitted
  • Kirk , B.S. , Peterson , J.W. , Stogner , R.M. and Carey , G.F. 2006 . libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations . Eng. Comput. , 23 : 237 – 254 .

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