Free Access
Issue
ESAIM: M2AN
Volume 39, Number 1, January-February 2005
Page(s) 157 - 181
DOI https://doi.org/10.1051/m2an:2005006
Published online 15 March 2005
  1. B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525–528. [Google Scholar]
  2. J.A. Atwell and B.B. King, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Mod. 33 (2001) 1–19. [Google Scholar]
  3. E. Balmes, Parametric families of reduced finite element models: Theory and applications. Mech. Syst. Signal Process. 10 (1996) 381–394. [Google Scholar]
  4. E. Balsa-Canto, A.A. Alonso and J.R. Banga, Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Indust. Engineering Chemistry Res. 43 (2004) 3353–3363. [CrossRef] [Google Scholar]
  5. H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser (1989). [Google Scholar]
  6. M. Barrault, N.C. Nguyen, Y. Maday and A.T. Patera, An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Sér. I. 339 (2004) 667–672. [Google Scholar]
  7. A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. [MathSciNet] [Google Scholar]
  8. R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In ENUMATH 95 Proc. World Sci. Publ., Singapore (1997). [Google Scholar]
  9. D. Bertsimas and J.N. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). [Google Scholar]
  10. E.A. Christensen, M. Brøns and J.N. Sørensen, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput. 21 (2000) 1419–1434. [CrossRef] [Google Scholar]
  11. W. Desch, F. Kappel and K. Kunisch, Eds., Control and Estimation of Distributed Parameter Systems, volume 126 of International Series of Numerical Mathematics. Birkhäuser (1998). [Google Scholar]
  12. N.H. El-Farra and P.D. Christofides, Coordinating feedback and switching for control of spatially distributed processes. Comput. Chemical Engineering 28 (2004) 111–128. [CrossRef] [Google Scholar]
  13. J.P. Fink and W.C. Rheinboldt, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21–28. [CrossRef] [MathSciNet] [Google Scholar]
  14. M. Grepl, Reduced-Basis Approximations for Time-Dependent Partial Differential Equations: Application to Optimal Control. Ph.D. Thesis, Massachusetts Institute of Technology (2005) (in progress). [Google Scholar]
  15. K.-H. Hoffmann, G. Leugering and F. Tröltzsch, Eds., Optimal Control of Partial Differential Equations, volume 133 of International Series of Numerical Mathematics. Birkhäuser (1998). [Google Scholar]
  16. K. Ito and S.S. Ravindran, A reduced basis method for control problems governed by PDEs, in Control and Estimation of Distributed Parameter Systems, W. Desch, F. Kappel, and K. Kunisch Eds., Birkhäuser (1998) 153–168. [Google Scholar]
  17. K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403–425. [CrossRef] [MathSciNet] [Google Scholar]
  18. K. Ito and S.S. Ravindran, Reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid Dyn. 15 (2001) 97–113. [Google Scholar]
  19. S. Lall, J.E. Marsden and S. Glavaski, A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12 (2002) 519–535. [CrossRef] [Google Scholar]
  20. M. Lin Lee, Estimation of the error in the reduced basis method solution of differential algebraic equation systems. SIAM J. Numer. Anal. 28 (1991) 512–528. [CrossRef] [MathSciNet] [Google Scholar]
  21. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971). [Google Scholar]
  22. L. Machiels, Y. Maday, I.B. Oliveira, A.T. Patera and D.V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Sér. I 331 (2000) 153–158. [Google Scholar]
  23. Y. Maday, A.T. Patera and D.V. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar Volume XIV, D. Cioranescu and J.-L. Lions Eds., Elsevier Science B.V. (2002) 533–569. [Google Scholar]
  24. M. Mattingly, E.A. Bailey, A.W. Dutton, R.B. Roemer and S. Devasia, Reduced-order modeling for hyperthermia: An extended balanced-realization-based approach. IEEE Transactions on Biomedical Engineering 45 (1998) 1154–1162. [CrossRef] [Google Scholar]
  25. M. Mattingly, R.B. Roemer and S. Devasia, Exact temperature tracking for hyperthermia: A model-based approach. IEEE Trans. Control Systems Technology 8 (2000) 979–992. [CrossRef] [Google Scholar]
  26. B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control 26 (1981) 17–32. [Google Scholar]
  27. D.A. Nagy, Modal representation of geometrically nonlinear behaviour by the finite element method. Comput. Structures 10 (1979) 683–688. [CrossRef] [Google Scholar]
  28. A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462. [CrossRef] [Google Scholar]
  29. I.B. Oliveira and A.T. Patera, Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Engineering (2005) (submitted). [Google Scholar]
  30. H.M. Park, T.Y. Yoon and O.Y. Kim, Optimal control of rapid thermal processing systems by empirical reduction of modes. Ind. Eng. Chem. Res. 38 (1999) 3964–3975. [CrossRef] [Google Scholar]
  31. J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. [CrossRef] [Google Scholar]
  32. T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487–496. [Google Scholar]
  33. T.A. Porsching and M. Lin Lee, The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24 (1987) 1277–1287. [CrossRef] [MathSciNet] [Google Scholar]
  34. C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering 124 (2002) 70–80. [Google Scholar]
  35. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997). [Google Scholar]
  36. S.S. Ravindaran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 34 (2000) 425–448. [Google Scholar]
  37. W.C. Rheinboldt, On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. 21 (1993) 849–858. [Google Scholar]
  38. D.V. Rovas, L. Machiels and Y. Maday, Reduced-basis output bound methods for parabolic problems. IMA J. Appl. Math. (2005) (submitted). [Google Scholar]
  39. D.V. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (2002). [Google Scholar]
  40. L. Sirovich and M. Kirby, Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Amer. A 4 (1987) 519–524. [Google Scholar]
  41. K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible navier-stokes equations; Rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Methods Fluids (2005) (to appear). [Google Scholar]
  42. K. Veroy, C. Prud'homme and A.T. Patera, Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris, Sér. I 337 (2003) 619–624. [Google Scholar]
  43. K. Veroy, C. Prud'homme, D.V. Rovas and A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (AIAA Paper 2003–3847), in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (June 2003). [Google Scholar]
  44. K. Veroy, D. Rovas and A.T. Patera, A Posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “Convex inverse" bound conditioners. ESAIM: COCV 8 (2002) 1007–1028. Special Volume: A tribute to J.-L. Lions. [CrossRef] [EDP Sciences] [Google Scholar]
  45. K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, in 15th AIAA Computational Fluid Dynamics Conference, AIAA (June 2001). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you