L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations


The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However this decomposition, usually implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.

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  1. 1.

    Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA J. 16(5), 525–528 (1978)

    Article  Google Scholar 

  2. 2.

    P. Astrid. Fast reduced order modeling technique for large scale ltv systems. In Proceedings of the 2004 American control conference, volume 1, pages 762–767. IEEE, 2004

  3. 3.

    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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    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. Math. 339(9), 667–672 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Benaceur, A., Ehrlacher, V., Ern, A., Meunier, S.: A progressive reduced basis/empirical interpolation method for nonlinear parabolic problems. SIAM J. Sci. Comput. 40(5), A2930–A2955 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    R. Bos, X. Bombois, and P. Van den Hof. Accelerating large-scale non-linear models for monitoring and control using spatial and temporal correlations. In Proceedings of the 2004 American Control Conference, volume 4, pages 3705–3710. IEEE, 2004

  8. 8.

    K. Carlberg, M. Barone, and H. Antil. Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction. J. Comput. Phys., 330:693–734, 2017

  9. 9.

    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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    F. Casenave, A. Ern, and T. Lelièvre. Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. ESAIM: M2AN, 48(1):207–229, 2014

  12. 12.

    Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Chen, Y., Gottlieb, S.: Reduced collocation methods: Reduced basis methods in the collocation framework. J. Sci. Comput. 55(3), 718–737 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Y. Chen, S. Gottlieb, and Y. Maday. Parametric analytical preconditioning and its applications to the reduced collocation methods. C. R. Acad. Sci. Paris, Ser. I, 352:661–666, 2014

  15. 15.

    Chen, Y., Hesthaven, J.S., Maday, Y., Rodriguez, J.: Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(2), 970–996 (2010)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Chen, Y., Hesthaven, J.S., Maday, Y., Rodriguez, J., Zhu, X.: Certified reduced basis method for electromagnetic scattering and radar cross section estimation. Comput. Methods Appl. Mech. Engrg. 233–236, 92–108 (2012)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Chen, Y., Jiang, J., Narayan, A.: A robust error estimator and a residual-free error indicator for reduced basis methods. Comput. Math. Appl. 77, 1963–1979 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Deparis, S., Rozza, G.: Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: applications to natural convection in a cavity. J. Comput. Phys. 228(12), 4359–4378 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Dimitriu, G., Ştefănescu, R., Navon, I.M.: Comparative numerical analysis using reduced-order modeling strategies for nonlinear large-scale systems. J. Comput. Appl. Math. 310, 32–43 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Everson, R., Sirovich, L.: Karhunen-loeve procedure for gappy data. J. Opt. Soci. Am. A 12(8), 1657–1664 (1995)

    Article  Google Scholar 

  21. 21.

    Fritzen, F., Haasdonk, B., Ryckelynck, D., Schöps, S.: An algorithmic comparison of the hyper-reduction and the discrete empirical interpolation method for a nonlinear thermal problem. Math. Comput. Appl. 23(1), 8 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Berkooz, P.H.G., Lumley, J.: The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25(1), 539–575 (1993)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Galbally, D., Fidkowski, K., Willcox, K., Ghattas, O.: Non-linear model reduction for uncertainty quantification in large-scale inverse problems. Int. J. Numer. Methods Eng. 81(12), 1581–1608 (2010)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Grepl, M.A.: Certified reduced basis methods for nonaffine linear time-varying and nonlinear parabolic partial differential equations. Math. Models Methods Appl. Sci. 22(03), 1150015 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    M. A. Grepl, Y. Maday, N. Nguyen, and A. T. Patera. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN, 41(3):575–605, 2007

  26. 26.

    M. A. Grepl and A. T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN, 39(1):157–181, 2005

  27. 27.

    B. Haasdonk. Chapter 2: Reduced basis methods for parametrized PDEs–a tutorial introduction for stationary and instationary problems, volume 15, pages 65–136. SIAM Philadelphia, 2017

  28. 28.

    B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN, 42(2):277–302, 2008

  29. 29.

    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics, Springer, Berlin (2016)

    Google Scholar 

  30. 30.

    Huynh, D.B.P., Knezevic, D.J., Chen, Y., Hesthaven, J.S., Patera, A.T.: A natural-norm Successive Constraint Method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199, 1963–1975 (2010)

    MathSciNet  Article  Google Scholar 

  31. 31.

    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 constants. C. R. Acad. Sci. Paris, Série I., 345:473–478, 2007

  32. 32.

    Ji, L., Chen, Y., Xu, Z.: A reduced basis method for the nonlinear Poisson-Boltzmann equation. Adv. Appl. Math. Mech. 11(5), 1200–1218 (2019)

    MathSciNet  Article  Google Scholar 

  33. 33.

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

    MathSciNet  Article  Google Scholar 

  34. 34.

    Liang, Y., Lee, H., Lim, S., Lin, W., Lee, K., Wu, C.: Proper orthogonal decomposition and its applications-Part I: Theory. J. Sound Vib. 252(3), 527–544 (2002)

    Article  Google Scholar 

  35. 35.

    Maday, Y., Mula, O., Turinici, G.: Convergence analysis of the generalized empirical interpolation method. SIAM J. Numer. Anal. 54(3), 1713–1731 (2016)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general multipurpose interpolation procedure: The magic points. Commun. Pure Appl. Anal. 8(1), 383 (2009)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)

    MathSciNet  Article  Google Scholar 

  38. 38.

    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)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Nguyen, N.C., Rozza, G., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo 46(3), 157–185 (2009)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980)

    Article  Google Scholar 

  41. 41.

    B. Peherstorfer. Sampling low-dimensional markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference. arXiv preprint arXiv:1908.11233, 2019

  42. 42.

    Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: An introduction. Springer Series in Computational Mathematics, Springer, New York (2015)

    Google Scholar 

  44. 44.

    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. Comput. Methods Eng. 15(3), 229–275 (2008)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Ryckelynck, D.: A priori hyperreduction method: an adaptive approach. J. Comput. Phys. 202(1), 346–366 (2005)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Ryckelynck, D.: Hyper-reduction of mechanical models involving internal variables. Int. J. Numer. Methods Eng. 77(1), 75–89 (2009)

    MathSciNet  Article  Google Scholar 

  47. 47.

    K. Veroy, C. Prud Homme, and A. T. Patera. Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Math., 337(9):619–624, 2003

  48. 48.

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

    Article  Google Scholar 

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Correspondence to Yanlai Chen.

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L. Ji and Z. Xu acknowledge the support from NSFC (grant Nos. 12071288 and 21773165), SSTC (grant No. 20JC1414100), and HPC center of Shanghai Jiao Tong University. Y. Chen was partially supported by National Science Foundation grant DMS-1719698 and by AFOSR grant FA9550-18-1-0383. L. Ji was partly supported by China Scholarship Council (CSC, No.201906230067) during the author’s one year visit at University of Massachusetts, Dartmouth. A. Narayan was partially supported by NSF award DMS-1848508. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and by the Simons Foundation Grant No. 50736 while Y. Chen, L. Ji, and A. Narayan were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Model and dimension reduction in uncertain and dynamic systems” program.

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Chen, Y., Ji, L., Narayan, A. et al. L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations. J Sci Comput 87, 10 (2021). https://doi.org/10.1007/s10915-021-01416-z

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