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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

  • Toni Lassila
  • Andrea Manzoni
  • Alfio Quarteroni
  • Gianluigi RozzaEmail author
Chapter
Part of the MS&A - Modeling, Simulation and Applications book series (MS&A, volume 9)

Abstract

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

Keywords

Proper Orthogonal Decomposition Proper Orthogonal Decomposition Mode Posteriori Error Estimation Reduce Basis Galerkin Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ahuja, MS., Rowley, C.: Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech 645, 447–478 (2010)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Akhtar, I., Nayfeh, A., Ribbens, C.: On the stability and extension of reduced-order Galerkin models in incompressible flows. Theor. Comp. Fluid Dyn. 23(3), 213–237 (2009)zbMATHGoogle Scholar
  3. 3.
    Amsallem, D., Cortial, J., Carlberg, K., Farhat, C.: A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Engr. 80(9), 1241–1258 (2009)zbMATHGoogle Scholar
  4. 4.
    Amsallem, D., Farhat, C.: An online method for interpolating linear parametric reducedorder models. SIAM J. Sci. Comput. 33(5), 2169–2198 (2011)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Amsallem, D., Zahr, M., Farhat, C.: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Engr. 92(10), 891–916 (2012)MathSciNetGoogle Scholar
  6. 6.
    Antil, H., Heinkenschloss, M., Hoppe, R.: Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optim. Methods Softw. 26(4-5), 643–669 (2011)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM (2005)Google Scholar
  8. 8.
    Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE. T. Automat. Contr. 53(10), 2237–2251 (2008)MathSciNetGoogle Scholar
  9. 9.
    Aubry, N., Holmes, P., Lumley, J., Stone, E.: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192(115), 173 (1988)Google Scholar
  10. 9a.
    Aubry, N., Holmes, P., Lumley, J., Stone, E.: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192(115), 355 (1988)Google Scholar
  11. 10.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)zbMATHMathSciNetGoogle Scholar
  12. 11.
    Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1971).zbMATHMathSciNetGoogle Scholar
  13. 12.
    Baiges, J., Codina, R., Idelsohn, S.: Explicit reduced order models for the stabilized finite element approximation of the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids (2013). DOI  10.1002/fld.3777
  14. 13.
    Barbagallo, A., Sipp, D., Schmid, P.J.: Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641(1), 1–50 (2009)zbMATHGoogle Scholar
  15. 14.
    Barone, M.F., Kalashnikova, I., Segalman, D.J., Thornquist, H.K.: Stable Galerkin reduced order models for linearized compressible flow. J. Comp. Phys. 228(6), 1932–1946 (2009)zbMATHMathSciNetGoogle Scholar
  16. 15.
    Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris. Sér. I Math. 339(9), 667–672 (2004)zbMATHMathSciNetGoogle Scholar
  17. 16.
    Bergmann, M., Bruneau, C., Iollo, A.: Enablers for robust POD models. J. Comp. Phys. 228(2), 516–538 (2009)zbMATHMathSciNetGoogle Scholar
  18. 17.
    Bergmann, M., Cordier, L.: Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. J. Comp. Phys. 227(16), 7813–7840 (2008)zbMATHMathSciNetGoogle Scholar
  19. 18.
    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43, 1457–1472 (2011)zbMATHMathSciNetGoogle Scholar
  20. 19.
    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)MathSciNetGoogle Scholar
  21. 20.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991)Google Scholar
  22. 21.
    Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)zbMATHMathSciNetGoogle Scholar
  23. 22.
    Buffa, A., Maday, Y., Patera, A., Prud’homme, C., Turinici, G.: A priori convergence of the greedy algorithm for the parametrized reduced basis. ESAIM Math. Modelling Numer. Anal. 46(3), 595–603 (2012)zbMATHMathSciNetGoogle Scholar
  24. 23.
    Bui-Thanh, T., Willcox, K., Ghattas, O.: Parametric reduced-order models for probabilistic analysis of unsteady aerodynamics applications. AIAA J. 46(10) (2008)Google Scholar
  25. 24.
    Bui-Thanh, T., Willcox, K., Ghattas, O., van Bloemen Waanders, B.: Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comp. Phys. 224(2), 880–896 (2007)zbMATHGoogle Scholar
  26. 25.
    Burkardt, J., Gunzburger, M., Lee, H.: Centroidal Voronoi tessellation-based reduced-order modeling of complex systems. SIAM J. Sci. Comput. 28(2), 459–484 (2006)zbMATHMathSciNetGoogle Scholar
  27. 26.
    Burkardt, J., Gunzburger, M., Lee, H.: POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Meth. Appl. Mech. Engrg. 196(1-3), 337–355 (2006)zbMATHMathSciNetGoogle Scholar
  28. 27.
    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., Amsterdam (1997)Google Scholar
  29. 28.
    Carlberg, K., Farhat, C.: A low-cost, goal-oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems. Int. J. Numer. Methods Engr. 86(3), 381–402 (2011)zbMATHMathSciNetGoogle Scholar
  30. 29.
    Cazemier, W., Verstappen, R., Veldman, A.: Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10, 1685 (1998)Google Scholar
  31. 30.
    Chaturantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)zbMATHMathSciNetGoogle Scholar
  32. 31.
    Chen, X., Akella, S., Navon, I.: A dual-weighted trust-region adaptive POD 4-D Var applied to a finite-volume shallow water equations model on the sphere. Int. J. Numer. Methods Fluids 68(3), 377–402 (2012)zbMATHMathSciNetGoogle Scholar
  33. 32.
    Chen, Y., Hesthaven, J., Maday, Y., Rodriguez, J.: A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris. Sér. I Math. 346, 1295–1300 (2008)zbMATHMathSciNetGoogle Scholar
  34. 33.
    Christensen, E., Brøns, M., Sørensen, J.: Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput. 21, 1419–1434 (2000)zbMATHGoogle Scholar
  35. 34.
    Cochelin, B., Vergez, C.: A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J. Sound Vibration 324(1), 243–262 (2009)Google Scholar
  36. 35.
    Colonius, T., Rowley, C., Freund, J., Murray, R.: On the choice of norm for modeling compressible flow dynamics at reduced-order using the POD. In: Proc. 41st IEEE Conf. on Decision and Control, vol. 3, pp. 3273–3278. IEEE (2002)Google Scholar
  37. 36.
    Daescu, D., Navon, I.: Efficiency of a POD-based reduced second-order adjoint model in 4D-Var data assimilation. Int. J. Numer. Methods Fluids 53(6), 985–1004 (2007)zbMATHMathSciNetGoogle Scholar
  38. 37.
    Daescu, D., Navon, I.: A dual-weighted approach to order reduction in 4DVAR data assimilation. Monthly Weather Review 136(3), 1026–1041 (2008)Google Scholar
  39. 38.
    Dahmen, W., Huang, C., Schwab, C., Welper, G.: Adaptive Petrov-Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50(5) (2012)Google Scholar
  40. 39.
    Deane, A., Kevrekidis, I., Karniadakis, G., Orszag, S.: Low-dimensionalmodels for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids 3(10), 2337–2354 (1991)zbMATHGoogle Scholar
  41. 40.
    Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov-Galerkin methods. part I: The transport equation. Comput. Methods Appl. Mech. Engr. 199(23-24), 1558–1572 (2010)zbMATHMathSciNetGoogle Scholar
  42. 41.
    Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov-Galerkin methods. II. optimal test functions. Numer. Methods Partial Differential Equations 27(1), 70–105 (2011)zbMATHMathSciNetGoogle Scholar
  43. 42.
    Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Num. Anal. 46(4), 2039–2067 (2008)zbMATHMathSciNetGoogle Scholar
  44. 43.
    Deparis, S., Løvgren, A.: Stabilized reduced basis approximation of incompressible three-dimensional Navier-Stokes equations in parametrized deformed domains. J. Sci. Comput. 50(1), 198–212 (2012)zbMATHMathSciNetGoogle Scholar
  45. 44.
    Deparis, S., Rozza, G.: Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity. J. Comp. Phys. 228(12), 4359–4378 (2009)zbMATHMathSciNetGoogle Scholar
  46. 45.
    Djouadi, S.: On the connection between balanced proper orthogonal decomposition, balanced truncation, and metric complexity theory for infinite dimensional systems. In: Proc. Am. Control Conf., June 30–July 2, Baltimore, MD, 2010 pp. 4911–4916 (2010)Google Scholar
  47. 46.
    Du, Q., Gunzburger, M.: Model reduction by proper orthogonal decomposition coupled with centroidal Voronoi tessellation. In: Proc. Fluids Engineering Division Summer Meeting, FEDSM2002-31051, ASME (2002)Google Scholar
  48. 47.
    Dumon, A., Allery, C., Ammar, A.: Proper general decomposition (PGD) for the resolution of Navier-Stokes equations. J. Comp. Phys. 230, 1387–1407 (2011)zbMATHMathSciNetGoogle Scholar
  49. 48.
    Eftang, J., Knezevic, D., Patera, A.: An “hp” certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dynam. Syst. 17(4), 395–422 (2011)zbMATHMathSciNetGoogle Scholar
  50. 49.
    Fahl, M.: Trust-region methods for flow control based on reduced order modelling. Ph.D. thesis, Universität Trier (2001)Google Scholar
  51. 50.
    Gerner, A., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comp. 34(5), A2812–A2836 (2012)zbMATHMathSciNetGoogle Scholar
  52. 51.
    Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations: Theory and algorithms. Springer-Verlag, Berlin Heidelberg New York (1986)zbMATHGoogle Scholar
  53. 52.
    Grepl, M., Patera, A.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM Math. Modelling Numer. Anal. 39(1), 157–181 (2005)zbMATHMathSciNetGoogle Scholar
  54. 53.
    Grinberg, L., Yakhot, A., Karniadakis, G.: Analyzing transient turbulence in a stenosed carotid artery by proper orthogonal decomposition. Ann. Biomed. Eng. 37(11), 2200–2217 (2009)Google Scholar
  55. 54.
    Gunzburger, M., Peterson, J., Shadid, J.: Reducer-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 196, 1030–1047 (2007)zbMATHMathSciNetGoogle Scholar
  56. 55.
    Haasdonk, B.: Convergence rates of the POD-greedy method. ESAIM Math. Modelling Numer. Anal. (2013). DOI  10.1051/m2an/2012045
  57. 56.
    Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM Math. Modelling Numer. Anal. 42(02), 277–302 (2008)zbMATHMathSciNetGoogle Scholar
  58. 57.
    Hall, K., Thomas, J., Clark, W.: Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA J. 40(5), 879–886 (2002)Google Scholar
  59. 58.
    Hay, A., Borggaard, J., Akhtar, I., Pelletier, D.: Reduced-order models for parameter dependent geometries based on shape sensitivity analysis. J. Comp. Phys. 229(4), 1327–1352 (2010)zbMATHMathSciNetGoogle Scholar
  60. 59.
    Hay, A., Borggaard, J., Pelletier, D.: Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition. J. Fluid Mech. 629, 41–72 (2009)zbMATHMathSciNetGoogle Scholar
  61. 60.
    Holmes, P., Lumley, J., Berkooz, G.: Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  62. 61.
    Huynh, D., Knezevic, D., Chen, Y., Hesthaven, J., Patera, A.: A natural-norm successive constraint method for inf-sup lower bounds. Comput. Meth. Appl. Mech. Engrg. 199(29-32), 1963–1975 (2010)zbMATHMathSciNetGoogle Scholar
  63. 62.
    Huynh, D., Knezevic, D., Patera, A.: A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM Math. Modelling Numer. Anal. 47(1), 213–251 (2013)zbMATHMathSciNetGoogle Scholar
  64. 63.
    Huynh, D., Knezevic, D., Peterson, J., Patera, A.: High-fidelity real-time simulation on deployed platforms. Comp. Fluids 43(1), 74–81 (2011)zbMATHMathSciNetGoogle Scholar
  65. 64.
    Huynh, D., Rozza, G., Sen, S., Patera, A.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability costants. C. R. Acad. Sci. Paris. Sér. I Math. 345, 473–478 (2007)zbMATHMathSciNetGoogle Scholar
  66. 65.
    Iapichino, L., Quarteroni, A., Rozza, G.: A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Engrg. 221-222, 63–82 (2012)MathSciNetGoogle Scholar
  67. 66.
    Iollo, A., Lanteri, S., Désidéri, J.: Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations. Theor. Comp. Fluid Dyn. 13(6), 377–396 (2000)zbMATHGoogle Scholar
  68. 67.
    Johansson, P., Andersson, H., Rønquist, E.: Reduced-basis modeling of turbulent plane channel flow. Comput. Fluids 35(2), 189–207 (2006)zbMATHGoogle Scholar
  69. 68.
    Johnson, C., Rannacher, R., Boman, M.: Numerics and hydrodynamic stability: toward error control in computational fluid dynamics. SIAM J. Numer. Anal. 32(4), 1058–1079 (1995)zbMATHMathSciNetGoogle Scholar
  70. 69.
    Kim, T.: Frequency-domain Karhunen-Loève method and its application to linear dynamic systems. AIAA J. 36(11), 2117–2123 (1998)Google Scholar
  71. 70.
    Knezevic, D., Nguyen, N., Patera, A.: Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations. Math. Mod. and Meth. in Appl. Sc. 21(7), 1415–1442 (2011)zbMATHMathSciNetGoogle Scholar
  72. 71.
    Knezevic, D.J.: Reduced basis approximation and a posteriori error estimates for a multiscale liquid crystal model. Math. Comput. Model. Dynam. Syst. 17(4), 443–461 (2011)zbMATHMathSciNetGoogle Scholar
  73. 72.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2003)MathSciNetGoogle Scholar
  74. 73.
    Kunisch, K., Volkwein, S.: Optimal snapshot location for computing POD basis functions. ESAIM Math. Modelling Numer. Anal. 44(3), 509 (2010)zbMATHMathSciNetGoogle Scholar
  75. 74.
    Lall, S., Marsden, J., Glavaški, S.: Empirical model reduction of controlled nonlinear systems. In: Proc. IFAC World Congress Vol. F. Int. Federation Automatic Control, Beijing, 1999 pp. 473–478 (1999)Google Scholar
  76. 75.
    Lall, S., Marsden, J., Glavaški, S.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12(6), 519–535 (2002)zbMATHGoogle Scholar
  77. 76.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: A reduced computational and geometrical framework for inverse problems in haemodynamics. Int. J. Numer. Meth. Biomed. Engng. 29(7), 741–776 (2013)MathSciNetGoogle Scholar
  78. 77.
    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Generalized reduced basis methods and n-width estimates for the approximation of the solutionmanifold of parametric PDEs. In: F. Brezzi, P. Colli Franzone, U. Gianazza, G. Gilardi (eds.) Analysis and Numerics of Partial Differential Equations. INdAM Series, vol. 4. Springer (2013). Re-printed also on Bollettino Unione Matematica Italiana (UMI), under permission/agreement Springer-UMIGoogle Scholar
  79. 78.
    Lassila, T., Quarteroni, A., Rozza, G.: A reduced basis model with parametric coupling for fluid-structure interaction problem. SIAM J. Sci. Comput. 34(2), A1187–A1213 (2012)zbMATHMathSciNetGoogle Scholar
  80. 79.
    Leblond, C., Allery, C., Inard, C.: An optimal projection method for the reduced-order modeling of incompressible flows. Comput. Methods Appl. Mech. Engrg. 200, 2507–2527 (2011)zbMATHMathSciNetGoogle Scholar
  81. 80.
    Lions, P.: Mathematical Topics in Fluid Mechanics. Oxford Lecture Series in Mathematics and Its Applications. Clarendon Press, Oxford, UK (1996)Google Scholar
  82. 81.
    Løvgren, A., Maday, Y., Rønquist, E.: A reduced basis element method for the steady Stokes problem. ESAIM Math. Modelling Numer. Anal. 40(3), 529–552 (2006)Google Scholar
  83. 82.
    Lucia, D., Beran, P., Silva, W.: Reduced-order modeling: new approaches for computational physics. Prog. Aerosp. Sci. 40(1-2), 51–117 (2004)Google Scholar
  84. 83.
    Lumley, J.: The structure of inhomogeneous turbulent flows. In: Yaglom, A.M., Tatarski, (eds.) Atmospheric turbulence and radio wave propagation pp. 166–178 (1967)Google Scholar
  85. 84.
    Ma, X., Karniadakis, G., Park, H., Gharib, M.: DPIV-driven flow simulation: a new computational paradigm. Proc. R. Soc. A 459(2031), 547–565 (2003)zbMATHMathSciNetGoogle Scholar
  86. 85.
    Maday, Y., Nguyen, N., Patera, A., Pau, G.: A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8(1) (2009)Google Scholar
  87. 86.
    Maday, Y., Patera, A., 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. Sér. I Math. 335, 1–6 (2002)MathSciNetGoogle Scholar
  88. 87.
    Manzoni, A.: Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Lausanne (2012)Google Scholar
  89. 88.
    Manzoni, A., Quarteroni, A., Rozza, G.: Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Methods Fluids 70(5), 646–670 (2012)MathSciNetGoogle Scholar
  90. 89.
    Maple, R., King, P., Orkwis, P., Wolff, J.: Adaptive harmonic balance method for nonlinear time-periodic flows. J. Comp. Phys. 193(2), 620–641 (2004)zbMATHGoogle Scholar
  91. 90.
    McMullen, M.: The application of non-linear frequency domain methods to the Euler and Navier-Stokes equations. Ph.D. thesis, Stanford University, Stanford, CA, USA (2003)Google Scholar
  92. 91.
    Moore, B.: Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Contr. 26(1) (1981)Google Scholar
  93. 92.
    Néron, D., Ladevèze, P.: Proper generalized decomposition for multiscale and multiphysics problems. Arch. Comput. Methods Engrg. 17(4), 351–372 (2010)zbMATHGoogle Scholar
  94. 93.
    Nguyen, N., Rozza, G., Patera, A.: Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation. Calcolo 46(3), 157–185 (2009)zbMATHMathSciNetGoogle Scholar
  95. 94.
    Nguyen, N., Veroy, K., Patera, A.: Certified real-time solution of parametrized partial differential equations. In: Yip, S. (Ed.). Handbook of Materials Modeling pp. 1523–1558 (2005)Google Scholar
  96. 95.
    Noack, B., Afanasiev, K., Morzynski, M., Tadmor, G., Thiele, F.: A hierarchy of lowdimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497(1), 335–363 (2003)zbMATHMathSciNetGoogle Scholar
  97. 96.
    Noack, B., Papas, P., Monkewitz, P.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523(1), 339–365 (2005)zbMATHMathSciNetGoogle Scholar
  98. 97.
    Noor, A., Peters, J.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980)Google Scholar
  99. 98.
    Peterson, J.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10, 777–786 (1989)zbMATHGoogle Scholar
  100. 99.
    Quarteroni, A.: Numerical Models for Differential Problems, 2nd ed. Modeling, Simulation and Applications (MS&A), Vol. 8. Springer-Verlag Italia, Milano (2014)Google Scholar
  101. 100.
    Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier-Stokes equations by reduced basis methods. Numer. Methods Partial Differential Equations 23(4), 923–948 (2007)zbMATHMathSciNetGoogle Scholar
  102. 101.
    Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1(3) (2011)Google Scholar
  103. 102.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin Heidelberg (1994)Google Scholar
  104. 103.
    Ravindran, S.: Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput. 15(4), 457–478 (2000)zbMATHMathSciNetGoogle Scholar
  105. 104.
    Ravindran, S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 34, 425–448 (2000)zbMATHMathSciNetGoogle Scholar
  106. 105.
    Rowley, C.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifur. Chaos Appl. Sci. Engrg. 15(3), 997–1014 (2005)zbMATHMathSciNetGoogle Scholar
  107. 106.
    Rozza, G., Huynh, D., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125(1), 741–776 (2013). DOI  10.1007/s00211-013-0534-8 MathSciNetGoogle Scholar
  108. 107.
    Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Engrg. 15, 229–275 (2008)zbMATHMathSciNetGoogle Scholar
  109. 108.
    Rozza, G., Manzoni, A., Negri, F.: Reduction strategies for PDE-constrained optimization problems in haemodynamics. In: J. Eberhardsteiner et.al. (ed.) Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, Austria, September 10-14, 2012 (2012)Google Scholar
  110. 109.
    Rozza, G., Veroy, K.: On the stability of reduced basis methods for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196(7), 1244–1260 (2007)zbMATHMathSciNetGoogle Scholar
  111. 110.
    Sen, S., Veroy, K., Huynh, P., Deparis, S., Nguyen, N., Patera, A.: “natural norm” a posteriori error estimators for reduced basis approximations. J. Comp. Phys. 217(1), 37–62 (2006)zbMATHMathSciNetGoogle Scholar
  112. 111.
    Sirisup, S., Karniadakis, G.: A spectral viscosity method for correcting the long-term behavior of POD models. J. Comp. Phys. 194(1), 92–116 (2004)zbMATHMathSciNetGoogle Scholar
  113. 112.
    Sirisup, S., Karniadakis, G.: Stability and accuracy of periodic flow solutions obtained by a POD-penalty method. J. Phys. D 202(3), 218–237 (2005)zbMATHMathSciNetGoogle Scholar
  114. 113.
    Sirisup, S., Karniadakis, G., Yang, Y., Rockwell, D.: Wave-structure interaction: simulation driven by quantitative imaging. Proc. R. Soc. A 460(2043), 729–755 (2004)zbMATHMathSciNetGoogle Scholar
  115. 114.
    Sirovich, L.: Turbulence and the dynamics of coherent structures. Part I: Coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)zbMATHMathSciNetGoogle Scholar
  116. 115.
    Tadmor, E.: Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal. 26(1), 30–44 (1989)zbMATHMathSciNetGoogle Scholar
  117. 116.
    Tadmor, G., Lehmann, O., Noack, B., Morzyński, M.: Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22, 034 (2010)Google Scholar
  118. 116a.
    Tadmor, G., Lehmann, O., Noack, B., Morzyński, M.: Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22, 102 (2010)Google Scholar
  119. 117.
    Tamellini, L., Le Maître, O., Nouy, A.: Model reduction based on proper generalized decomposition for the stochastic steady incompressible Navier-Stokes equations. Tech. Rep. 26, MOX — Modellistica e calcolo scientifico, Politecnico di Milano (2012)Google Scholar
  120. 118.
    Temam, R.: Navier-Stokes Equations. AMS Chelsea, Providence, Rhode Island (2001)Google Scholar
  121. 119.
    Terragni, F., Vega, J.M.: On the use of POD-based ROMs to analyze bifurcations in some dissipative systems. Physica D: Nonlinear Phenomena 241(17), 1393–1405 (2012). DOI  10.1016/j.physd.2012.04.009 zbMATHGoogle Scholar
  122. 120.
    Thomas, J., Hall, K., Dowell, E.: A harmonic balance approach for modeling nonlinear aeroelastic behavior of wings in transonic viscous flow. In: 44 th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference (2003)Google Scholar
  123. 121.
    Tonn, T., Urban, K., Volkwein, S.: Optimal control of parameter-dependent convectiondiffusion problems around rigid bodies. SIAM J. Sci. Comput. 32(3), 1237–1260 (2010)zbMATHMathSciNetGoogle Scholar
  124. 122.
    Tu, J., Rowley, C.: An improved algorithm for balanced POD through an analytic treatment of impulse response tails. J. Comp. Phys. 231(16) (2012)Google Scholar
  125. 123.
    Urban, K., Patera, A.T.: An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comp. (2013). In pressGoogle Scholar
  126. 124.
    Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Meth. Fluids 47(8-9), 773–788 (2005)zbMATHMathSciNetGoogle Scholar
  127. 125.
    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 16thAIAAComputational Fluid Dynamics Conference (2003). Paper 2003-3847Google Scholar
  128. 126.
    Wang, Z., Akhtar, I., Borggaard, J., Iliescu, T.: Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison. Comput. Meth. Appl. Mech. Engrg. 237-240, 10–26 (2012)MathSciNetGoogle Scholar
  129. 127.
    Weller, J., Lombardi, E., Bergmann, M., Iollo, A.: Numerical methods for low-order modeling of fluid flows based on POD. Int. J. Numer. Methods Fluids 63(2), 249–268 (2010)zbMATHMathSciNetGoogle Scholar
  130. 128.
    Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40(11), 2323–2330 (2002)Google Scholar
  131. 129.
    Yano, M.: A space-time Petrov-Galerkin certified reduced basis method: application to the Boussinesq equations. Submitted to SIAM J. Scientific Computing (revised, 2013). Preprint available at augustine.mit.edu
  132. 130.
    Yano, M., Patera, A.T.: A space-time variational approach to hydrodynamic stability theory. Proceedings of Royal Society A, 469(2155), article 2013 0036 (2013)Google Scholar
  133. 131.
    Yano, M., Patera, A.T., Urban, K.: A space-time certified reduced basis method for Burgers’ equation (2013). Preprint available at augustine.mit.edu
  134. 132.
    Zhou, K., Salomon, G., Wu, E.: Balanced realization and model reduction for unstable systems. Int. J. Robust Nonlinear Control 9(3), 183–198 (1999)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Toni Lassila
    • 1
  • Andrea Manzoni
    • 2
  • Alfio Quarteroni
    • 1
    • 3
  • Gianluigi Rozza
    • 2
    Email author
  1. 1.MATHICSE-CMCS Modelling and Scientific ComputingEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.SISSA Mathlab — International School for Advanced StudiesTriesteItaly
  3. 3.MOX — Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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