Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 76)


Reduced basis approximations for geometrically parametrized advection-diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.


  1. 1.
    M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris, 339(9):667–672, 2004MATHMathSciNetGoogle Scholar
  2. 2.
    Y. Chen, J. Hestaven, Y. Maday, and J. Rodriguez. A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris, Ser. I, 346:1295–1300, 2008Google Scholar
  3. 3.
    L. Dedè. Adaptive and reduced basis methods for optimal control problems in environmental applications. PhD thesis, Politecnico di Milano, 2008Google Scholar
  4. 4.
    D.B.P. Huynh, D. Knezevic, Y. Chen, J. Hestaven, and A.T. Patera. A natural-norm successive constraint method for inf-sub lower bounds. Scientific Computing Group, Brown University, No. 2009–23Google Scholar
  5. 5.
    D.B.P. Huynh, N.C. Nguyen, G. Rozza, and A.T. Patera. Rapid reliable solution of the parametrized partial differential equations of continuum mechanics and transport, 2008.
  6. 6.
    D.B.P Huynh, G. Rozza, S. Sen, and A.T. Patera. 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, 2007Google Scholar
  7. 7.
    T. Lassila and G. Rozza. Parametric free-form shape design with PDE models and reduced basis method. Comput. Meth. Appl. Mech. Eng., 199(23–24):1583–1592, 2010CrossRefMathSciNetGoogle Scholar
  8. 8.
    N.C. Nguyen. A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations. J. Comp. Phys., 227:983–1006, 2007MATHCrossRefGoogle Scholar
  9. 9.
    A.T. Patera and G. Rozza. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering., 2009. Available at
  10. 10.
    A. Quarteroni, G. Rozza, and A. Quaini. Reduced basis methods for optimal control of advection-diffusion problem. In Advances in Numerical Mathematics, W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, and Y. Vassilevski, Editors, pages 193–216, 2007Google Scholar
  11. 11.
    A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer, Berlin, 2008MATHGoogle Scholar
  12. 12.
    G. Rozza. Reduced basis methods for Stokes equations in domains with non-affine parameter dependence. Comput. Vis. Sci., 12(1):23–35, 2009CrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng., 15:229–275, 2008MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    T.W. Sederberg and S.R. Parry. Free-form deformation of solid geometric models. Comput. Graph., 20(4):151–160, 1986CrossRefGoogle Scholar
  15. 15.
    T. Tonn and K. Urban. A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. In Proc. ECCOMAS CFD, 2006Google Scholar
  16. 16.
    K. Veroy, C. Prud’homme, D.V. Rovas, and A.T. Patera. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proc. 16th AIAA Comput. Fluid Dyn., 2003Google Scholar

Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.MATHICSE-CMCS Modelling and Scientific ComputingÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Institute of MathematicsHelsinki University of TechnologyHelsinkiFinland

Personalised recommendations