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Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

Application to Transport and Continuum Mechanics

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Abstract

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; a posteriori error estimation procedures—rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies—minimum marginal 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 (2007) http://augustine.mit.edu/methodology/methodology_rbMIT_System.htm. MIT, Cambridge

  2. Ainsworth M, Oden JT (1997) A posteriori error estimation in finite element analysis. Comput Methods Appl Mech Eng 142:1–88

    MATH  MathSciNet  Google Scholar 

  3. Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Wiley-Interscience, New York

    MATH  Google Scholar 

  4. Almroth BO, Stern P, Brogan FA (1978) Automatic choice of global shape functions in structural analysis. AIAA J 16:525–528

    Google Scholar 

  5. Anderson TL (2005) Fracture mechanics: fundamentals and application, 3rd edn. CRC Press, Boca Raton

    Google Scholar 

  6. Arpaci VS (1966) Conduction heat transfer. Addison-Wesley, Reading

    MATH  Google Scholar 

  7. Arpaci VS, Larsen PS (1984) Convection heat transfer. Prentice Hall, Englewood Cliffs

    Google Scholar 

  8. Atwell JA, King BB (2001) Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math Comput Model 33(1–3):1–19

    MATH  MathSciNet  Google Scholar 

  9. Babuška I (1971) Error-bounds for finite element method. Numer Math 16:322–333

    MATH  MathSciNet  Google Scholar 

  10. Babuška I, Osborn J (1991) Eigenvalue problems. In: Handbook of numerical analysis, vol II. Elsevier, Amsterdam, pp 641–787

    Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. Babuška I, Strouboulis T (2001) The finite element method and its reliability. Numerical mathematics and scientific computation. Clarendon, Oxford

    Google Scholar 

  14. Bai ZJ (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl Numer Math 43(1–2):9–44

    MATH  MathSciNet  Google Scholar 

  15. Balmes E (1996) Parametric families of reduced finite element models: Theory and applications. Mech Syst Signal Process 10(4):381–394

    Google Scholar 

  16. Balsa-Canto E, Alonso A, Banga J (2004) Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Ind Eng Chem Res 43(13):3353–3363

    Google Scholar 

  17. Banks HT, Kunisch K (1989) Estimation techniques for distributed parameter systems. Systems & control: foundations & applications. Birkhäuser, Boston

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. Barsom JM, Rolfe ST (1999) Fracture and fatigue control in structures. American society for testing and metals. Butterworth, Stoneham

    Google Scholar 

  21. Bashir O, Willcox K, Ghattas O, var Bloemen Waanders B, Hill J (2008) Hessian-based model reduction for large-scale systems with initial condition inputs. Int J Numer Methods Eng 73(6):844–868

    Google Scholar 

  22. Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. Benner P, Mehrmann V, Sorensen D (eds) (2003) Dimension reduction of large-scale systems. Lecture notes in computational science and engineering. Springer, Heidelberg

    Google Scholar 

  25. Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis of periodic structures. North-Holland, Amsterdam

    Google Scholar 

  26. Boyaval S (2007) Application of reduced basis approximation and a posteriori error estimation to homogenization theory. Multiscale Model Simul (to appear)

  27. Braess D (2001) Finite elements. Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  29. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer series in computational mathematics, vol 15. Springer, Berlin

    MATH  Google Scholar 

  30. Brezzi F, Rappaz J, Raviart P (1980) Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer Math 36:1–25

    MATH  MathSciNet  Google Scholar 

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

  32. Bui-Thanh T, Willcox K, Ghattas O (2008) Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J Sci Comput (to appear)

  33. Bui-Thanh T, Willcox K, Ghattas O (2007) 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

  34. Caloz G, Rappaz J (1997) Numerical analysis for nonlinear and bifurcation problems. In: Ciarlet P, Lions J (eds) Handbook of numerical analysis. Techniques of scientific computing (Part 2), vol V. Elsevier, Amsterdam, pp 487–637

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  36. Cancès E, Le Bris C, Nguyen NC, Maday Y, Patera AT, Pau GSH (2007) 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)

  37. Cazemier W (1997) Proper orthogonal decomposition and low dimensional models for turbolent flows. University of Groningen, Groningen

    Google Scholar 

  38. Chen J, Kang SM (2001) 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

  39. Chen Y, White J (2000) A quadratic method for nonlinear model order reduction. In: Proceedings of the international conference on modeling and simulation of microsystems, pp 477–480

  40. Christensen E, Brøns M, Sørensen J (2000) Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J Sci Comput 21(4):1419–1434

    MATH  Google Scholar 

  41. Ciarlet PG (2002) The finite element method for elliptic problems. Classics in applied mathematics, vol 40. SIAM, Philadelphia

    Google Scholar 

  42. Daniel L, Ong C, White J (2002) Geometrically parametrized interconnect performance models for interconnect synthesis. In: Proceedings of the 2002 international symposium on physical design. Assoc Comput Mach, New York, pp 202–207

    Google Scholar 

  43. Dedè L (2008) Advanced numerical methods for the solution of optimal control problems described by pdes with environmental applications. PhD thesis, Politecnico di Milano

  44. Demmel JW (1997) Applied numerical linear algebra. SIAM, Philadelphia

    MATH  Google Scholar 

  45. Deparis S (2008) Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J Numer Anal 46:2039

    MathSciNet  MATH  Google Scholar 

  46. Farle O, Hill V, Nickel P, Dyczij-Edlinger R (2006) Multivariate finite element model order reduction for permittivity or permeability estimation. IEEE Trans Megn 42:623–626

    Google Scholar 

  47. Fink JP, Rheinboldt WC (1983) On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z Angew Math Mech 63(1):21–28

    MATH  MathSciNet  Google Scholar 

  48. Fox RL, Miura H (1971) An approximate analysis technique for design calculations. AIAA J 9(1):177–179

    Google Scholar 

  49. Ganapathysubramanian S, Zabaras N (2004) Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Comput Methods Appl Mech Eng 193:5017–5034

    MATH  Google Scholar 

  50. Girault V, Raviart P (1986) Finite element approximation of the Navier-Stokes equations. Springer, Berlin

    Google Scholar 

  51. Goberna MA, Lopez MA (1998) Linear semi-infinite optimization. Wiley, New York

    MATH  Google Scholar 

  52. Grepl M (2005) Reduced-basis approximations and a posteriori error estimation for parabolic partial differential equations. PhD thesis, Massachusetts Institute of Technology

  53. Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Model Math Anal Numer

  54. Grepl MA, Nguyen NC, Veroy K, Patera AT, Liu GR (2007) Certified rapid solution of partial differential equations for real-time parameter estimation and optimization. In: Biegler LT, Ghattas O, Heinkenschloss M, Keyes D, van Wandeers B (eds) Proceedings of the 2nd Sandia workshop of PDE-constrained optimization: Real-time PDE-constrained optimization. SIAM computational science and engineering book series. SIAM, Philadelphia, pp 197–216

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  56. Gresho P, Sani R (1998) Incompressible flow and the finite element method: advection-diffusion and isothermal laminar flow. Wiley, New York

    MATH  Google Scholar 

  57. Gunzburger MD (1989) Finite element methods for viscous incompressible flows. Academic Press, San Diego

    MATH  Google Scholar 

  58. Gunzburger MD (2003) Perspectives in flow control and optimization. Advances in design and control. SIAM, Philadelphia

    MATH  Google Scholar 

  59. Gunzburger MD, Peterson J, Shadid JN (2007) Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput Methods Appl Mech 196:1030–1047

    MATH  MathSciNet  Google Scholar 

  60. Haasdonk B, Ohlberger M (2008) Reduced basis method for finite volume approximations of parametrized evolution equations. Math Model Numer Anal 42(2):277–302

    MATH  MathSciNet  Google Scholar 

  61. Huynh DBP (2007) Reduced-basis approximation and application to fracture and inverse problems. PhD thesis, Singapore-MIT Alliance, National University of Singapore

  62. Huynh DBP, Patera AT (2007) Reduced-basis approximation and a posteriori error estimation for stress intensity factors. Int J Numer Methods Eng 72(10):1219–1259

    Google Scholar 

  63. Huynh DBP, Peraire J, Patera AT, Liu GR (2007) Reduced basis approximation and a posteriori error estimation for stress intensity factors: Application to failure analysis. In: Singapore-MIT alliance symposium

  64. Huynh DBP, Rozza G, Sen S, Patera AT (2007) A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C R Acad Sci Paris Ser I 345:473–478

    MATH  MathSciNet  Google Scholar 

  65. Isaacson E, Keller HB (1994) Computation of eigenvalues and eigenvectors, analysis of numerical methods. Dover, New York

    Google Scholar 

  66. Ito K, Ravindran SS (1998) A reduced basis method for control problems governed by PDEs. In: Desch W, Kappel F, Kunisch K (eds) Control and estimation of distributed parameter systems. Birkhäuser, Boston, pp 153–168

    Google Scholar 

  67. Ito K, Ravindran SS (1998) A reduced-order method for simulation and control of fluid flows. J Comput Phys 143(2):403–425

    MATH  MathSciNet  Google Scholar 

  68. Ito K, Ravindran SS (2001) Reduced basis method for optimal control of unsteady viscous flows. Int J Comput Fluid Dyn 15(2):97–113

    MATH  MathSciNet  Google Scholar 

  69. Ito K, Schroeter JD (2001) Reduced order feedback synthesis for viscous incompressible flows. Math Comput Model 33(1–3):173–192

    MATH  MathSciNet  Google Scholar 

  70. Jabbar M, Azeman A (2004) Fast optimization of electromagnetic-problems: the reduced-basis finite element approach. IEEE Trans Magn 40(4):2161–2163

    Google Scholar 

  71. Johnson CR (1989) A Gershgorin-type lower bound for the smallest singular value. Linear Algebra Appl 112:1–7

    MATH  MathSciNet  Google Scholar 

  72. Karhunen K (1946) Zur spektraltheorie stochastischer prozesse. Ann Acad Sci Fenn 37

  73. Kunisch K, Volkwein S (2002) Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Numer Anal 40(2):492–515

    MATH  MathSciNet  Google Scholar 

  74. Kwang ATY (2006) 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

  75. Le Bris C (2006) Private communication. MIT

  76. Lee MYL (1991) Estimation of the error in the reduced-basis method solution of differential algebraic equations. SIAM J Numer Anal 28:512–528

    MATH  MathSciNet  Google Scholar 

  77. LeGresley PA, Alonso JJ (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 conference and exhibit. Denver, CO (2000). Paper 2000-2545

  78. Loeve MM (1955) Probability theory. Van Nostrand, Princeton

    MATH  Google Scholar 

  79. Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for complex flow systems. In: Wesseling P, Oñate E, Periaux J (eds) ECCOMAS CFD 2006 proceedings. TU Delft, Delft

    Google Scholar 

  80. Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for the steady Stokes problem. Math Model Numer Anal 40(3):529–552

    MathSciNet  Google Scholar 

  81. Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems. Model Identif Control 27(2):79–94

    MathSciNet  Google Scholar 

  82. Løvgren AE, Maday Y, Rønquist EM (2007) The reduced basis element method for fluid flows. In: Analysis and simulation of fluid dynamics. Advances in mathematical fluid mechanics. Birkauser, Boston, pp 129–154

    Google Scholar 

  83. Ly H, Tran H (2001) Modeling and control of physical processes using proper orthogonal decomposition. Math Comput Model 33:223–236

    MATH  Google Scholar 

  84. Machiels L, Maday Y, Oliveira IB, Patera A, Rovas D (2000) Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C R Acad Sci Paris, Sér I 331(2):153–158

    MATH  MathSciNet  Google Scholar 

  85. Maday Y, Patera A, Turinici G (2002) A Priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J Sci Comput 17(1–4):437–446

    MATH  MathSciNet  Google Scholar 

  86. Maday Y, Patera AT, Rovas DV (2002) A blackbox reduced-basis output bound method for noncoercive linear problems. In: Cioranescu D, Lions JL (eds) Nonlinear partial differential equations and their applications, Collége de France Seminar, vol XIV. Elsevier, Amsterdam, pp 533–569

    Google Scholar 

  87. Maday Y, Patera AT, Turinici G (2002) Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C R Acad Sci Paris, Sér I 335(3):289–294

    MATH  MathSciNet  Google Scholar 

  88. Meyer CD (2000) Matrix analysis and applied linear algebra. SIAM, Philadelphia

    MATH  Google Scholar 

  89. Meyer M, Matthies HG (2003) Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comput Mech 31(1–2):179–191

    MATH  Google Scholar 

  90. Mortenson ME (1990) Computer graphics handbook. Industrial Press, New York

    Google Scholar 

  91. Murakami Y (2001) Stress intensity factors handbook. Elsevier, Amsterdam

    Google Scholar 

  92. Nagy DA (1979) Modal representation of geometrically nonlinear behaviour by the finite element method. Comput Struct 10:683–688

    MATH  Google Scholar 

  93. Newman AJ (1996) Model reduction via the Karhunen-Loeve expansion part i: an exposition. Technical report, Institute for System Research University of Maryland, pp 96–322

  94. Newman JN (1977) Marine hydrodynamics. MIT Press, Cambridge

    Google Scholar 

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

  96. Nguyen NC, Patera AT (2007) Efficient and reliable parameter estimation in heat conduction using Bayesian inference and a reduced basis method (in preparation)

  97. Nguyen NC, Veroy K, Patera AT (2005) Certified real-time solution of parametrized partial differential equations. In: Yip S (ed) Handbook of materials modeling. Springer, Berlin, pp 1523–1558

    Google Scholar 

  98. Noor AK (1981) Recent advances in reduction methods for nonlinear problems. Comput Struct 13:31–44

    MATH  Google Scholar 

  99. Noor AK (1982) On making large nonlinear problems small. Comput Methods Appl Mech Eng 34:955–985

    MATH  Google Scholar 

  100. Noor AK, Balch CD, Shibut MA (1984) Reduction methods for non-linear steady-state thermal analysis. Int J Numer Methods Eng 20:1323–1348

    MATH  Google Scholar 

  101. Noor AK, Peters JM (1980) Reduced basis technique for nonlinear analysis of structures. AIAA J 18(4):455–462

    Google Scholar 

  102. Noor AK, Peters JM (1981) Bifurcation and post-buckling analysis of laminated composite plates via reduced basis techniques. Comput Methods Appl Mech Eng 29:271–295

    MATH  Google Scholar 

  103. Noor AK, Peters JM (1981) Tracing post-limit-point paths with reduced basis technique. Comput Methods Appl Mech Eng 28:217–240

    MATH  Google Scholar 

  104. Noor AK, Peters JM (1983) Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates. Int J Numer Methods Eng 19:1783–1803

    MATH  Google Scholar 

  105. Noor AK, Peters JM (1983) Recent advances in reduction methods for instability analysis of structures. Comput Struct 16:67–80

    MATH  Google Scholar 

  106. Noor AK, Peters JM, Andersen CM (1984) Mixed models and reduction techniques for large-rotation nonlinear problems. Comput Methods Appl Mech Eng 44:67–89

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  109. Parks DM (1974) A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fract 10(4):487–502

    MathSciNet  Google Scholar 

  110. Parlett BN (1998) The symmetric eigenvalue problem. SIAM, Philadelphia

    MATH  Google Scholar 

  111. Patera AT, Rønquist EM (2007) Reduced basis approximations and a posteriori error estimation for a Boltzmann model. Comput Methods Appl Mech Eng 196:2925–2942

    MATH  Google Scholar 

  112. Patera AT, Rozza G (2008) Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Copyright MIT (2006–2007). MIT Pappalardo monographs in mechanical engineering (to appear)

  113. Pau GSH (2007) Reduced-basis method for quantum models of periodic solids. PhD thesis, Massachusetts Institute of Technology

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

    MATH  Google Scholar 

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

  116. Phillips JR (2003) Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems. IEEE Trans Comput Aided Des Integr Circuits Syst 22:171–187

    Google Scholar 

  117. Pierce N, Giles MB (2000) Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev 42(2):247–264

    MathSciNet  Google Scholar 

  118. Pironneau O (2006) Calibration of barrier options. In: Fitzgibbon W, Hoppe R, Periaux J, Pironneau O, Vassilevski Y (eds) Advances in numerical mathematics. Moscow/Houston, Russian Academy of Sciences/University of Houston, pp 183–192

    Google Scholar 

  119. Porsching TA (1985) Estimation of the error in the reduced basis method solution of nonlinear equations. Math Comput 45(172):487–496

    MATH  MathSciNet  Google Scholar 

  120. Porsching TA, Lee MYL (1987) The reduced-basis method for initial value problems. SIAM J Numer Anal 24:1277–1287

    MATH  MathSciNet  Google Scholar 

  121. Prud’homme C, Rovas D, Veroy K, Maday Y, Patera A, Turinici G (2002) Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. J Fluids Eng 124(1):70–80

    Google Scholar 

  122. Prud’homme C, Rovas D, Veroy K, Patera AT (2002) A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. Model Math Anal Numer 36(5):747–771

    MATH  MathSciNet  Google Scholar 

  123. Quarteroni A, Rozza G (2007) Numerical solution of parametrized Navier-Stokes equations by reduced basis method. Numer Methods Partial Differ Equ 23:923–948

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  125. Quarteroni A, Sacco R, Saleri F (2000) Numerical mathematics. Texts in applied mathematics, vol 37. Springer, New York

    Google Scholar 

  126. Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations, 2nd edn. Springer, Berlin

    Google Scholar 

  127. Ravindran SS (2000) Reduced-order adaptive controllers for fluid flows using pod. J Sci Comput 15(4):457–478

    MATH  MathSciNet  Google Scholar 

  128. Ravindran SS (2000) A reduced order approach to optimal control of fluids flow using proper orthogonal decomposition. Int J Numer Methods Fluids 34(5):425–448

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  130. Rewienski M, White J (2003) A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans Comput-Aided Des Integr Circuits Syst 22:155–170

    Google Scholar 

  131. Rheinboldt WC (1981) Numerical analysis of continuation methods for nonlinear structural problems. Comput Struct 13(1–3):103–113

    MATH  MathSciNet  Google Scholar 

  132. Rheinboldt WC (1993) On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal Theory, Methods Appl 21(11):849–858

    MATH  MathSciNet  Google Scholar 

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

  134. Rovas DV (2002) Reduced-basis output bound methods for parametrized partial differential equations. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA

  135. Rozza G (2005) Real-time reduced basis techniques for arterial bypass geometries. In: Bathe K (ed) Proceedings of the third MIT conference on computational fluid and solid mechanics, June 14–17, 2005. Computational fluid and solid mechanics. Elsevier, Amsterdam, pp 1283–1287

    Google Scholar 

  136. Rozza G (2005) Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity. Appl Numer Math 55(4):403–424

    MATH  MathSciNet  Google Scholar 

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

  138. Rozza G (2008) Reduced basis method for Stokes equations in domains with non-affine parametric dependence. Comput Vis Sci 11(4). doi:10.1007/s00791-006-0044-7

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

    MathSciNet  Google Scholar 

  140. Schiesser WE, Silebi CA (1997) Computational transport phenomena: numerical methods for the solution of transport problems. Cambridge University Press, Cambridge

    Google Scholar 

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

  142. Sen S, Veroy K, Huynh DBP, Deparis S, Nguyen NC, Patera AT (2006) “Natural norm” a posteriori error estimators for reduced basis approximations. J Comput Phys 217:37–62

    MATH  MathSciNet  Google Scholar 

  143. Shi G, Shi CJR (2004) 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, New York, pp 774–779

    Google Scholar 

  144. Sirisup S, Xiu D, Karniadakis G (2005) Equation-free/Galerkin-free POD-assisted computation of incompressible flows. J Comput Phys 207:617–642

    MathSciNet  Google Scholar 

  145. Sirovich L (1987) Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Q Appl Math 45(3):561–571

    MATH  MathSciNet  Google Scholar 

  146. Strang G (2003) Introduction to linear algebra. Wellesley-Cambridge, Wellesley

    Google Scholar 

  147. Strang G, Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, New York

    MATH  Google Scholar 

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

    Google Scholar 

  149. Trefethen L, III DB (1997) Numerical linear algebra. SIAM, Philadelphia

    MATH  Google Scholar 

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

  151. Veroy K, Patera AT (2005) Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; Rigorous reduced-basis a posteriori error bounds. Int J Numer Methods Fluids 47:773–788

    MATH  MathSciNet  Google Scholar 

  152. Veroy K, Prud’homme C, Patera AT (2003) Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C R Acad Sci Paris, Sér I 337(9):619–624

    MATH  MathSciNet  Google Scholar 

  153. Veroy K, Prud’homme C, Rovas DV, Patera AT (2003) 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. Paper 2003-3847

  154. Wang J, Zabaras N (2005) Using Bayesian statistics in the estimation of heat source in radiation. Int J Heat Mass Transfer 48:15–29

    MATH  Google Scholar 

  155. Weile DS, Michielssen E (2001) Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction. IEEE Trans Antennas Propag 49(11):1539–1549

    MATH  MathSciNet  Google Scholar 

  156. Weile DS, Michielssen E, Gallivan K (2001) Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Trans Antennas Propag 49(5):801–813

    Google Scholar 

  157. Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40(11):2323–2330

    Article  Google Scholar 

  158. Zienkiewicz O, Taylor R (2000) Finite element method. The basis, vol 1. Butterworth-Heinemann, London

    Google Scholar 

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Authors and Affiliations

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

    G. Rozza

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

    D. B. P. Huynh

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

    A. T. Patera

Authors
  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 of Paris 6 and Luca Dedé of MOX-Politecnico di Milano.

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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. Arch Computat Methods Eng 15, 229–275 (2008). https://doi.org/10.1007/s11831-008-9019-9

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  • DOI: https://doi.org/10.1007/s11831-008-9019-9

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