Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws

  • Masayuki YanoEmail author


We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs) associated with steady hyperbolic and convection-dominated conservation laws. Our formulation builds on three ingredients: a discontinuous Galerkin (DG) method which provides stability for conservation laws, reduced basis (RB) spaces which provide low-dimensional approximations of the parametric solution manifold, and the empirical quadrature procedure (EQP) which provides hyperreduction of the Galerkin-projection-based reduced model. The hyperreduced system inherits the stability of the DG discretization: (i) energy stability for linear hyperbolic systems, (ii) symmetry and non-negativity for steady linear diffusion systems, and hence (iii) energy stability for linear convection-diffusion systems. In addition, the framework provides (a) a direct quantitative control of the solution error induced by the hyperreduction, (b) efficient and simple hyperreduction posed as a 1 minimization problem, and (c) systematic identification of the reduced bases and the empirical quadrature rule by a greedy algorithm. We demonstrate the formulation for parametrized aerodynamics problems governed by the compressible Euler and Navier-Stokes equations.


Parametrized nonlinear PDEs Conservation laws Model reduction Hyperreduction Empirical quadrature Discontinuous Galerkin method 

Mathematics Subject Classification (2010)

65N15 65N30 35Q35 76G25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We would like to thank Prof. Anthony Patera (MIT) for many fruitful discussions and the anonymous reviewers for their helpful feedback. We acknowledge the computational resources provided by Compute Canada/SciNet.

Funding information

This study was financially supported by the Natural Sciences and Engineering Research Council of Canada.


  1. 1.
    An, S.S., Kim, T., James, D.L.: Optimizing cubature for efficient integration of subspace deformations. ACM Trans. Graph. 27(5), 165:1–165:10 (2008). CrossRefGoogle Scholar
  2. 2.
    Antonietti, P.F., Pacciarini, P., Quarteroni, A.: A discontinuous Galerkin reduced basis element method for elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis 50(2), 337–360 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptical problems. SIAM J. Numer. Anal. 39 (5), 1749–1779 (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barone, M.F., Kalashnikova, I., Segalman, D.J., Thornquist, H.K.: Stable Galerkin reduced order models for linearized compressible flow. J. Comput. Phys. 228(6), 1932–1946 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Ser. I 339, 667–672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: Kroner, D., Olhberger, M., Rohde, C. (eds.) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp 195–282. Springer-Verlag (1999)Google Scholar
  8. 8.
    Bassi, F., Shu, S.: GMRES discontinuous Galerkin solution of the compressible Navier-stokes equations. In: Cockburn, K. (ed.) Discontinuous Galerkin Methods: Theory, Computation and Applications, pp 197–208. Springer, Berlin (2000)Google Scholar
  9. 9.
    Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. Part I: branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bui-Thanh, T., Murali, D., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics. AIAA 2003-4213, AIAA (2003)Google Scholar
  11. 11.
    Carlberg, K., Bou-Mosleh, C., Farhat, C.: Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86(2), 155–181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cockburn, B.: Discontinuous Galerkin methods. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fü,r Angewandte Mathematik und Mechanik 83(11), 731–754 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Everson, R., Sirovich, L.: Karhunen-Loève procedure for gappy data. J. Opt. Soc. Am. A 12(8), 1657–1664 (1995)CrossRefGoogle Scholar
  15. 15.
    Farhat, C., Avery, P., Chapman, T., Cortial, J.: Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int. J. Numer. Methods Eng. 98(9), 625–662 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Farhat, C., Chapman, T., Avery, P.: Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int. J. Numer. Methods Eng. 102(5), 1077–1110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41(3), 575–605 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hernández, J., Caicedo, M., Ferrer, A.: Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Comput. Methods Appl. Mesh. Eng. 313, 687–722 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Iollo, A., Lombardi, D.: Advection modes by optimal mass transfer. Physical Review E 89(2) (2014)Google Scholar
  22. 22.
    LeGresley, P.A., Alonso, J.J.: Investigation of non-linear projection for POD based reduced order models for aerodynamics. AIAA 2001–0926, AIAA (2001)Google Scholar
  23. 23.
    LeGresley, P.A., Alonso, J.J.: Dynamic domain decomposition and error correction for reduced order models. In: 41St Aerospace Sciences Meeting and Exhibit. American Institute of Aeronautics and Astronautics (2003)Google Scholar
  24. 24.
    Nguyen, N.C., Peraire, J.: An efficient reduced-order modeling approach for non-linear parametrized partial differential equations. Int. J. Numer. Methods Eng. 76(1), 27–55 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ohlberger, M., Rave, S.: Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. C.R. Math. 351(23-24), 901–906 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ohlberger, M., Rave, S.: Reduced basis methods: success, limitations and future challenges. In: Proceedings of the Conference Algoritmy, pp 1–12 (2016)Google Scholar
  27. 27.
    Ohlberger, M., Schindler, F.: Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Commun. 37(6), A2865–A2895 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Patera, A.T., Yano, M.: An LP empirical quadrature procedure for parametrized functions. C. R. Acad. Sci. Paris, Ser I (2017)Google Scholar
  29. 29.
    Pietro, D.A.D., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  30. 30.
    Pinkus, A: n-widths of Sobolev spaces in l p. Constr. Approx. 1(1), 15–62 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  32. 32.
    Riviere, B.M.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. The Society for Industrial and Applied Mathematics, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    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 — application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ryu, E.K., Boyd, S.P.: Extensions of gauss quadrature via linear programming. Found Comput. Math 15(4), 953–971 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Washabaugh, K., Amsallem, D., Zahr, M., Farhat, C.: Nonlinear model reduction for CFD problems using local reduced-order bases. AIAA 2012-2686, AIAA (2012)Google Scholar
  36. 36.
    Washabaugh, K., Zahr, M.J., Farhat, C.: On the use of discrete nonlinear reduced-order models for the prediction of steady-state flows past parametrically deformed complex geometries. AIAA 2016-1814 AIAA (2016)Google Scholar
  37. 37.
    Welper, G.: Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM J. Sci. Comput. 39(4), A1225–A1250 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yano, M., Modisette, J.M., Darmofal, D.: The importance of mesh adaptation for higher-order discretizations of aerodynamic flows. AIAA 2011-3852, AIAA (2011)Google Scholar
  39. 39.
    Yano, M., Patera, A.T.: An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs. Comput. Methods Appl. Mesh Eng. 344, 1104–1123 (2019)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zimmermann, R., Görtz, S.: Non-linear reduced order models for steady aerodynamics. Procedia Comput. Sci. 1(1), 165–174 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of TorontoInstitute for Aerospace StudiesTorontoCanada

Personalised recommendations