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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

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Abstract

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

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References

  1. rbMIT Software: http://augustine.mit.edu/methodology/ methodology rbMIT System.htm. © MIT, Cambridge, MA (2007)

  2. Ainsworth, M., Oden, J.T.:A posteriori error estimation in finite element analysis. Comp. Meth. Appl. Mech. Engrg.142, 1–88 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience (2000)

  4. Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA Journal16, 525–528 (1978)

    Article  Google Scholar 

  5. Anderson, T.L.: Fracture Mechanics: Fundamentals and Application, third edn. CRC (2005)

  6. Arpaci, V.S.: Conduction heat transfer. Addison-Wesley (1966)

  7. Arpaci, V.S., Larsen, P.S.: Convection heat transfer. Prentice Hall (1984)

  8. Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Mathematical and Computer Modelling33(1-3), 1–19 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Babuška, I.: Error-bounds for finite element method. Numerische Mathematik16, 322–333 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  10. Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol. II, pp. 641-787. Elsevier (1991)

  11. Babuška, I., Rheinboldt, W.:A posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng.12, 1597–1615 (1978)

    Article  MATH  Google Scholar 

  12. Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, 736–754 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  13. Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford,UK (2001)

    Google Scholar 

  14. Bai, Z.J.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Applied Numerical Mathematics43(1-2), 9–44 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Balmes, E.: Parametric families of reduced finite element models: Theory and applications. Mechanical Systems and Signal Processing10(4), 381–394 (1996)

    Article  Google Scholar 

  16. Balsa-Canto, E., Alonso, A., Banga, J.: Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Industrial & Engineering Chemistry Research43(13), 3353–3363 (2004)

    Article  Google Scholar 

  17. Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser (1989)

    MATH  Google Scholar 

  18. Barrault, M., Nguyen, N.C., Maday, Y., Patera, A.T.: An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Série I.339, 667–672 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Barrett, A., Reddien, G.: On the reduced basis method. Z. Angew. Math. Mech.75(7), 543–549 (1995)

    MATH  MathSciNet  Google Scholar 

  20. Barsom, J.M., Rolfe, S.T.: Fracture and Fatigue Control in Structures. American Society for Testing and Metals (1999)

  21. Bashir, O., Willcox, K., Ghattas, O.: Hessian-based model reduction for large-scale systems with initial condition inputs. Int. J. for Num. Meth. in Engineering (2007). Submitted

  22. Bathe, K.J.: Finite Element Procedures. Prentice Hall (1996)

  23. Becker, R., Rannacher, R.: A feedback approach to error control in finite element method: Basic analysis and examples. East - West J. Numer. Math.4, 237–264 (1996)

    MATH  MathSciNet  Google Scholar 

  24. Benner, P., Mehrmann, V., (Eds.), D.S.: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering. Springer, Heildeberg (2003)

    Google Scholar 

  25. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis of Periodic Structures. North-Holland, Amsterdam (1978)

    Google Scholar 

  26. Boyaval, S.: Application of reduced basis approximation and a posteriori error estimation to homogenization theory. SIAM Multiscale Modeling and Simulation (2007). Submitted

  27. Braess, D.: Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, UK (2001)

    MATH  Google Scholar 

  28. Brezzi, F.: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O., Anal. Numér.2, 129–151 (1974)

    MathSciNet  Google Scholar 

  29. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods,Springer Series in Computational Mathematics, vol. 15. Springer Verlag (1991)

  30. Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numerische Mathematik36, 1–25 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  31. Bui-Thanh, T., Damodaran, M., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics (AIAA Paper 2003-4213). In: Proceedings of the 15th AIAA Computational Fluid Dynamics Conference (2003)

  32. Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space (AIAA Paper 2007-2049). In: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference (2007)

  33. Caloz, G., Rappaz, J.: Numerical analysis for nonlinear and bifurcation problems. In: P. Ciarlet, J. Lions (eds.) Handbook of Numerical Analysis, Vol. V, Techniques of Scientific Computing (Part 2), pp. 487-637. Elsevier Science B.V. (1997)

  34. Cancès, E., Le Bris, C., Maday, Y., Turinici, G.: Towards reduced basis approaches inab initio electronic structure computations. J. Sci. Comput.17(1-4), 461–469 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Cancès, E., Le Bris, C., Nguyen, N.C., Maday, Y., Patera, A.T., Pau, G.S.H.: Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry. In: Proceedings of the Workshop for High-dimensional Partial Differential Equations in Science and Engineering (Montreal) (2007)

  36. Cazemier, W.: Proper Orthogonal Decomposition and Low Dimensional Models for Turbolent Flows. University of Groningen (1997)

  37. Chen, J., Kang, S.M.: Model-order reduction of nonlinear mems devices through arclength-based Karhunen-Loéve decomposition. In: Proceeding of the IEEE international Symposium on Circuits and Systems, vol. 2, pp. 457-460 (2001)

  38. Chen, Y., White, J.: A quadratic method for nonlinear model order reduction. In: Proceedings of the international Conference on Modeling and Simulation of Microsystems, pp. 477-480 (2000)

  39. Christensen, E., Brøns, M., Sørensen, J.: Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing21(4), 1419–1434 (2000)

    Article  MATH  Google Scholar 

  40. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, 40. SIAM (2002)

  41. Daniel, L., Ong, C., White, J.: Geometrically parametrized interconnect performance models for interconnect synthesis. In: Proceedings of the 2002 International Symposium on Physical Design, ACM press, pp. 202-207 (2002)

  42. Dedè, L.: Advanced numerical methods for the solution of optimal control problems described by pdes with environmental applications. Ph.D. thesis, Politecnico di Milano (In progress)

  43. Demmel, J.W.: Applied Numerical Linear Algebra. SIAM (1997)

  44. Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM Journal of Numerical Analysis (2007). Submitted

  45. Farle, O., Hill, V., Nickel, P., Dyczij-Edlinger, R.: Multivariate finite element model order reduction for permittivity or permeability estimation. IEEE Transactions on Megnetics42, 623–626 (2006)

    Article  Google Scholar 

  46. Fink, J.P., Rheinboldt, W.C.: On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech.63(1), 21–28 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  47. Fox, R.L., Miura, H.: An approximate analysis technique for design calculations. AIAA Journal9(1), 177–179 (1971)

    Article  Google Scholar 

  48. Ganapathysubramanian, S., Zabaras, N.: Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Computer Methods in Applied Mechanics and Engineering193, 5017–5034 (2004)

    Article  MATH  Google Scholar 

  49. Girault, V., Raviart, P.: Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag (1986)

  50. Goberna, M.A., Lopez, M.A.: Linear Semi-Infinite Optimization. J.Wiley, New York (1998)

    MATH  Google Scholar 

  51. Grepl, M.: Reduced-basis approximations and aposteriori error estimation for parabolic partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology (2005)

  52. Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. M2AN (Math. Model. Numer. Anal.) (2007)

  53. Grepl, M.A., Nguyen, N.C., Veroy, K., Patera, A.T., Liu, G.R.: Certified rapid solution of partial differential equations for real-time parameter estimation and optimization. In: L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, B. van B. Wandeers (eds.) Proceedings of the 2nd Sandia Workshop of PDE-Constrained Optimization: Real-Time PDE-Constrained Optimization, SIAM Computational Science and Engineering Book Series, pp. 197-216 (2007)

  54. Grepl, M.A., Patera, A.T.:A Posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN (Math. Model. Numer. Anal.)39(1), 157–181 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  55. Gresho, P., Sani, R.: Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow. John Wiley & Sons (1998)

  56. Gunzburger, M.D.: Finite Element Methods for Viscous Incompressible Flows. Academic Press (1989)

  57. Gunzburger, M.D.: Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM (2003)

  58. Gunzburger, M.D., Peterson, J., Shadid, J.N.: Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comp. Meth. Applied Mech.196, 1030–1047 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  59. Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized evolution equations. Mathematical Modelling and Numerical Analysis (M2AN) (2006). Submitted

  60. Huynh, D.B.P.: Reduced-basis approximation and application to fracture and inverse problems. Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore (2007)

    Google Scholar 

  61. Huynh, D.B.P., Patera, A.T.: Reduced-basis approximation anda posteriori error estimation for stress intensity factors. Int. J. Num. Meth. Eng. (2007). In press (DOI: 10.1002/nme.2090)

  62. Huynh, D.B.P., Peraire, J., Patera, A.T., Liu, G.R.: Reduced basis approximation and a posteriori error estimation for stress intensity factors: Application to failure analysis. In: Singapore-MIT Alliance Symposium (2007)

  63. Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci. Paris, Analyse Numérique (2007). Submitted

  64. Isaacson, E., Keller, H.B.: Computation of Eigenvalues and Eigenvectors, Analysis of Numerical Methods. Dover Publications, New York (1994)

    Google Scholar 

  65. Ito, K., Ravindran, S.S.: A reduced basis method for control problems governed by PDEs. In: W. Desch, F. Kappel, K. Kunisch (eds.) Control and Estimation of Distributed Parameter Systems, pp. 153-168. Birkhäuser (1998)

  66. Ito, K., Ravindran, S.S.: A reduced-order method for simulation and control of fluid flows. Journal of Computational Physics143(2), 403–425 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  67. Ito, K., Ravindran, S.S.: Reduced basis method for optimal control of unsteady viscous flows. International Journal of Computational Fluid Dynamics15(2), 97–113 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  68. Ito, K., Schroeter, J.D.: Reduced order feedback synthesis for viscous incompressible flows. Mathematical And Computer Modelling33(1-3), 173–192 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  69. Jabbar, M., Azeman, A.: Fast optimization of electromagnetic-problems:the reduced-basis finite element approach. IEEE Transactions on Magnetics40(4), 2161–2163 (2004)

    Article  Google Scholar 

  70. Johnson, C.R.: A Gershgorin-type lower bound for the smallest singular value. Linear Algebra and Appl112, 1–7 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  71. Karhunen, K.: Zur spektraltheorie stochastischer prozesse. Annales Academiae Scientiarum Fennicae 37 (1946)

  72. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Num. Analysis40(2), 492–515 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  73. Kwang, A.T.Y.: Reduced basis methods for 2nd order wave equation: Application to one dimensional seismic problem. Master’s thesis, Singapore-MIT Alliance, Computation for Design and Optimization (2006)

  74. Le Bris, C.: Private Communication. MIT (2006)

  75. Lee, M.Y.L.: Estimation of the error in the reducedbasis method solution of differential algebraic equations. SIAM Journal of Numerical Analysis28, 512–528 (1991)

    Article  MATH  Google Scholar 

  76. LeGresley, P.A., Alonso, J.J.: Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 Conference and Exhibit, Denver, CO (2000). Paper 2000-2545

  77. Loeve, M.M.: Probability Theory. Van Nostrand (1955)

  78. Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for complex flow systems. In: ECCOMAS CFD 2006 Proceedings, P. Wesseling, E. Oñate, J. Periaux (Eds.) TU Delft, The Netherlands (2006)

    Google Scholar 

  79. Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem. Mathematical Modelling and Numerical Analysis (M2AN)40(3), 529–552 (2006)

    Article  Google Scholar 

  80. Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems. Modeling, Identification and Control27(2), 79–94 (2006)

    Article  MathSciNet  Google Scholar 

  81. 82. Løvgren, A.E., Maday, Y., Rønquist, E.M.: The reduced basis element method for fluid flows. Journal of Mathematical Fluid Mechanics (2007). In press

  82. Ly, H., Tran, H.: Modeling and control of physical processes using proper orthogonal decomposition. Mathematical and Computer Modelling33,223–2366 (2001)

    Article  MATH  Google Scholar 

  83. Machiels, L., Maday, Y., Oliveira, I.B., Patera, A., Rovas, D.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Série I331(2), 153–158 (2000)

    MATH  MathSciNet  Google Scholar 

  84. Maday, Y., Patera, A., Turinici, G.: A Priori convergence theory for reduced-basis approximations of singleparameter elliptic partial differential equations. Journal of Scientific Computing17(1-4), 437–446 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  85. Maday, Y., Patera, A.T., Rovas, D.V.: A blackbox reduced-basis output bound method for noncoercive linear problems. In: D. Cioranescu, J.L. Lions (eds.) Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar Volume XIV, pp. 533-569. Elsevier Science B.V. (2002)

  86. Maday, Y., Patera, A.T., Turinici, G.: Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris, Série I335(3), 289–294 (2002)

    MATH  MathSciNet  Google Scholar 

  87. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM (2000)

  88. Meyer, M., Matthies, H.G.: Efficient model reduction in non-linear dynamics using the Karhunen-Léve expansion and dual-weighted-residual methods. Computational Mechanics31(1-2), 179–191 (2003)

    Article  MATH  Google Scholar 

  89. Mortenson, M.E.: Computer graphics handbook. Industrial Press (1990)

  90. Murakami, Y.: Stress Intensity Factors Handbook. Elsevier (2001)

  91. Nagy, D.A.: Modal representation of geometrically nonlinear behaviour by the finite element method. Computers and Structures10, 683–688 (1979)

    Article  MATH  Google Scholar 

  92. Newman, A.J.: Model reduction via the Karhunen-Loeve expansion part i: an exposition. Technical Report Institute for System Research University of Maryland (96-322) (1996)

  93. Newman, J.N.: Marine Hydrodynamics. MIT Press, Cambridge, MA (1977)

    Google Scholar 

  94. Nguyen, N.C.: Reduced-basis approximation and a posteriori error bounds for nonaffine and nonlinear partial differential equations: Application to inverse analysis. Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore (2005)

  95. Nguyen, N.C., Patera, A.T.: Efficient and reliable parameter estimation in heat conduction using Bayesian inference and a reduced basis method (2007). In preparation

  96. Nguyen, N.C., Veroy, K., Patera, A.T.: Certified realtime solution of parametrized partial differential equations. In: S. Yip (ed.) Handbook of Materials Modeling, pp. 1523-1558. Springer (2005)

  97. Noor, A.K.: Recent advances in reduction methods for nonlinear problems. Comput. Struct.13, 31–44 (1981)

    Article  MATH  Google Scholar 

  98. Noor, A.K.: On making large nonlinear problems small. Comp. Meth. Appl. Mech. Engrg.34, 955–985 (1982)

    Article  MATH  Google Scholar 

  99. Noor, A.K., Balch, C.D., Shibut, M.A.: Reduction methods for non-linear steady-state thermal analysis. Int. J. Num. Meth. Engrg.20, 1323–1348 (1984)

    Article  MATH  Google Scholar 

  100. Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA Journal18(4), 455–462 (1980)

    Article  Google Scholar 

  101. Noor, A.K., Peters, J.M.: Bifurcation and post-buckling analysis of laminated composite plates via reduced basis techniques. Comp. Meth. Appl. Mech. Engrg.29, 271–295 (1981)

    Article  MATH  Google Scholar 

  102. Noor, A.K., Peters, J.M.: Tracing post-limit-point paths with reduced basis technique. Comp. Meth. Appl. Mech. Engrg.28, 217–240 (1981)

    Article  MATH  Google Scholar 

  103. Noor, A.K., Peters, J.M.: Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates. Int. J. Num. Meth. Engrg.19, 1783–1803 (1983)

    Article  MATH  Google Scholar 

  104. Noor, A.K., Peters, J.M.: Recent advances in reduction methods for instability analysis of structures. Comput. Struct.16, 67–80 (1983)

    Article  MATH  Google Scholar 

  105. Noor, A.K., Peters, J.M., Andersen, C.M.: Mixed models and reduction techniques for large-rotation nonlinear problems. Comp. Meth. Appl. Mech. Engrg.44, 67–89 (1984)

    Article  MATH  Google Scholar 

  106. Oliveira, I., Patera, A.T.: Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng.8, 43–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  107. Paraschivoiu, M., Peraire, J., Maday, Y., Patera, A.T.: Fast bounds for outputs of partial differential equations. In: J. Borgaard, J. Burns, E. Cliff, S. Schreck (eds.) Computational methods for optimal design and control, pp. 323-360. Birkhäuser (1998)

  108. Parks, D.M.: A stiffness derivative finite element technique for determination of crack tip stress intensity factors. International Journal of Fracture10(4), 487–502 (1974)

    Article  MathSciNet  Google Scholar 

  109. Parlett, B.N.: The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics, Philadelphia (1998)

  110. Patera, A.T., Rønquist, E.M.: Reduced basis approximations and a posteriori error estimation for a Boltzmann model. Computer Methods in Applied Mechanics and Engineering196, 2925–2942 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  111. Patera, A.T., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Copyright MIT (2006-2007). To appear in MIT Pappalardo Monographs in Mechanical Engineering

  112. Pau, G.S.H.: Reduced-basis method for quantum models of periodic solids. Ph.D. thesis, Massachusetts Institute of Technology. (2007)

  113. Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10(4), 777–786 (1989)

    Article  MATH  Google Scholar 

  114. Phillips, J.R.: Projection frameworks for model reduction of weakly nonlinear systems. In: Proceeding of the 37th ACM/IEEE Design Automation Conference, pp. 184-189 (2000)

  115. Phillips, J.R.: Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems. In: IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems, vol. 22, pp. 171-187 (2003)

  116. Pierce, N., Giles, M.B.: Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Review42(2),247–2644 (2000)

    Article  MathSciNet  Google Scholar 

  117. Pironneau, O.: Calibration of barrier options. In: W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski (eds.) Advances in Numerical Mathematics, pp. 183–192. Moscow, Institute of Numerical Mathematics, Russian Academy of Sciences and Houston, Department of Mathematics, University of Houston (2006)

    Google Scholar 

  118. Porsching, T.A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation45(172), 487–496 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  119. Porsching, T.A., Lee, M.Y.L.: The reduced-basis method for initial value problems. SIAM Journal of Numerical Analysis24, 1277–1287 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  120. Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. Journal of Fluids Engineering124(1), 70–80 (2002)

    Article  Google Scholar 

  121. Prud’homme, C., Rovas, D., Veroy, K., Patera, A.T.: A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. M2AN Math. Model. Numer. Anal.36(5), 747–771 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  122. Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier-Stokes equations by reduced basis method. Num. Meth. PDEs23, 923–948 (2007)

    MATH  MathSciNet  Google Scholar 

  123. Quarteroni, A., Rozza, G., Quaini, A.: Reduced basis method for optimal control af advection-diffusion processes. In: W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski (eds.) Advances in Numerical Mathematics, pp. 193-216. Moscow, Institute of Numerical Mathematics, Russian Academy of Sciences and Houston, Department of Mathematics, University of Houston (2006)

  124. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, Texts in Applied Mathematics, vol. 37. Springer, New York (2000)

    Google Scholar 

  125. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, 2nd edn. Springer (1997)

  126. Ravindran, S.S.: Reduced-order adaptive controllers for fluid flows using pod. J. of Scientific Computing15(4), 457–478 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  127. Ravindran, S.S.: A reduced order approach to optimal control of fluids flow using proper orthogonal decomposition. Int. J. of Numerical Methods in Fluids34(5), 425–448 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  128. Ravindran, S.S.: Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput.23(6), 1924–1942 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  129. Rewienski, M., White, J.: A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. In: IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems, vol. 22, pp. 155-170 (2003)

  130. Rheinboldt, W.C.: Numerical analysis of continuation methods for nonlinear structural problems. Computers and Structures13(1-3), 103–113 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  131. Rheinboldt, W.C.: On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Analysis, Theory, Methods and Applications21(11), 849–858 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  132. Rovas, D., Machiels, L., Maday, Y.: Reduced basis output bounds methods for parabolic problems. IMA J. Appl. Math. (2005)

  133. Rovas, D.V.: Reduced-basis output bound methods for parametrized partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2002)

    Google Scholar 

  134. Rozza, G.: Real-time reduced basis techniques for arterial bypass geometries. In: K. Bathe (ed.) Computational Fluid and Solid Mechanics, pp. 1283-1287. Elsevier (2005). Proceedings of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005

  135. Rozza, G.: Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. Ph.D. thesis, EPFL, Ecole Polytechnique Federale de Lausanne (2005)

  136. Rozza, G.: Reduced basis method for Stokes equations in domains with non-affine parametric dependence. Comp. Vis. Science (2007). In press (DOI: 10.1007/s00791-006-0044-7)

  137. Rozza, G.: Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity. Appl. Numer. Math.55(4), 403–424 (2007)

    Article  MathSciNet  Google Scholar 

  138. Rozza, G., Veroy, K.: On the stability of reduced basis method for Stokes equations in parametrized domains. Comp. Meth. Appl. Mech. and Eng.196, 1244–1260 (2007)

    Article  MathSciNet  Google Scholar 

  139. Schiesser, W.E., Silebi, C.A.: Computational Transport Phenomena: Numerical Methods for the Solution of Transport Problems. Cambridge Univeristy Press (1997)

  140. Sen, S.: Reduced-basis approximation and a posteriori error estimation for non-coercive eliptic problems: Application to acoustics. Ph.D. thesis, Massachusetts Institute of Technology. (2007)

  141. Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C., Patera, A.T.: ‘Natural norm’ a posteriori error estimators for reduced basis approximations. Journal of Computational Physics217, 37–62 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  142. Shi, G., Shi, C.J.R.: Parametric model order reduction for interconnect analysis. In: Proceedings of the 2004 Conference on Asia South Pacific design automation: electronic design and solution fair, IEEE press, pp. 774-779 (2004)

  143. Sirisup, S., Xiu, D., Karniadakis, G.: Equationfree/ Galerkin-free POD-assisted computation of incompressible flows. Journal of Computational Physics207, 617–642 (2005)

    Article  MathSciNet  Google Scholar 

  144. Sirovich, L.: Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quarterly of Applied Mathematics45(3), 561–571 (1987)

    MATH  MathSciNet  Google Scholar 

  145. Strang, G.: Introduction to linear algebra. Wellesley-Cambridge Press (2003)

  146. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall (1973)

  147. Tonn, T., Urban, K.: A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. In: ECCOMAS CFD 2006 Proceedings, P. Wesseling, E. Oñate, J. Periaux (Eds.) TU Delft, The Netherlands (2006)

    Google Scholar 

  148. Trefethen, L., III, D.B.: Numerical Linear Algebra. SIAM (1997)

  149. Veroy, K.: Reduced-basis methods applied to problems in elasticity: Analysis and applications. Ph.D. thesis, Massachusetts Institute of Technology (2003)

  150. Veroy, K., Patera, A.T.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; Rigorous reduced-basisa posteriori error bounds. International Journal for Numerical Methods in Fluids47, 773–788 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  151. Veroy, K., Prud’homme, C., Patera, A.T.: Reducedbasis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris, Série I337(9), 619–624 (2003)

    MATH  MathSciNet  Google Scholar 

  152. Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A Posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (2003). Paper 2003-3847

  153. Wang, J., Zabaras, N.: Using Bayesian statistics in the estimation of heat source in radiation. International Journal of Heat and Mass Transfer48, 15–29 (2005)

    Article  MATH  Google Scholar 

  154. Weile, D.S., Michielssen, E.: Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction. IEEE Transactions on Antennas and Propagation49(11), 1539–1549 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  155. Weile, D.S., Michielssen, E., Gallivan, K.: Reducedorder modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Transactions on Antennas and Propagation49(5), 801–813 (2001)

    Article  Google Scholar 

  156. Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA Journal40(11), 2323–2330 (2002)

    Article  Google Scholar 

  157. Zienkiewicz, O., Taylor, R.: Finite Element Method: Volume 1. The Basis. Butterworth-Heinemann, London (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Mechanical Engineering Department,, Room 3-264, 77 Mass Avenue, 02142-4307, Cambridge, MA, USA

    G. Rozza

  2. National University of Singapore, Singapore-MIT Alliance, E4-04-10,4 Eng.Drive, 117576, Singapore

    D. B. P. Huynh

  3. Massachusetts Institute of Technology, Room 3-266,77 Mass Avenue, 02142-4307, Cambridge, MA, USA

    A. T. Patera

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  1. G. Rozza
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  2. D. B. P. Huynh
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  3. A. T. Patera
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Correspondence to G. Rozza.

Additional information

This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425,the Singapore-MIT Alliance,the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund,and the Progetto Roberto Rocca Politecnico di Milano-MIT.We acknowledge many helpful discussions with Professor Yvon Maday of University Paris6.

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Rozza, G., Huynh, D.B.P. & Patera, A.T. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. ARCO 15, 1–47 (2007). https://doi.org/10.1007/BF03024948

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