Skip to main content
SpringerLink
Log in

Adaptive sampling and modal expansions in pattern-forming systems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which are selected via Gauss elimination with double pivoting on the original snapshot matrix containing the given data. In the present work, the sampling method is adapted and combined with low-dimensional modeling. This combination yields a novel adaptive algorithm for the simulation of time-dependent non-linear dynamics in pattern-forming systems. Convenient snapshot sets, computed on demand over the evolution, are stored to record local temporal events whose underlying mechanisms are essential for the approximations. Also, a collection of sparse grid points, which are used to construct the mode basis and the reduced system of equations, is adaptively sampled according to unlinked spatial structures. The outcome is a reduced order model of the problem that (i) yields reliable approximations of the dynamical transitions, (ii) is well-suited to describe localized spatio-temporal complexity, and (iii) provides fast computations. Robustness, accuracy, and computational efficiency of the proposed algorithm are illustrated for some relevant pattern-forming systems, in both one and two spatial dimensions, exhibiting solutions with a rich spatio-temporal structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adrover, A., Giona, M.: Modal reduction of PDE models by means of Snapshot Archetypes. Physica D 182, 23–45 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akkari, N., Hamdouni, A., Liberge, E., Jazar, M.: A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow. J. Comput. Appl. Math. 270, 522–530 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alonso, D., Vega, J.M., Velazquez, A.: Reduced order model for viscous aerodynamic flow past an airfoil. AIAA J. 48, 1946–1958 (2010)

    Article  Google Scholar 

  4. Alonso, D., Vega, J.M., Velazquez, A., de Pablo, V.: Reduced-order modeling of three-dimensional external aerodynamic flows. J. Aerospace Eng. 25, 588–599 (2012)

    Article  Google Scholar 

  5. Amsallem, D., Zahar, M.J., Washabaugh, K.: Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction. Adv. Comput. Math. 41, 1187–1230 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aranson, I.S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 100–142 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation methods in models described by proper orthogonal decomposition. IEEE T. Automat. Contr. 53, 2237–2250 (2008)

    Article  MATH  Google Scholar 

  8. Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, Inc., New York (1989)

    MATH  Google Scholar 

  9. Bache, E., Vega, J.M., Velazquez, A.: Model reduction in the back step fluid-thermal problem with variable geometry. Int. J. Therm. Sci. 49, 2376–2384 (2010)

    Article  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bekemeyer, P., Ripepi, M., Heinrich, R., Görtz, S.: Nonlinear unsteady reduced-order modeling for gust-load predictions. AIAA J. 57, 1–12 (2019)

    Article  Google Scholar 

  12. Bergmann, M., Bruneau, C.H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228, 516–538 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Braconnier, T., Ferrier, M., Jouhaud, J.C., Montagnac, M., Sagaut, P.: Towards an adaptive POD/SVD surrogate model for aeronautic design. Comput. Fluids 40, 195–209 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cebeci, T.: Convective Heat Transfer. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  16. Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. India 78, 808–817 (2000)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Couplet, M., Basdevant, C., Sagaut, P.: Calibrated reduced-order POD-Galerkin system for fluid flow modelling. J. Comput. Phys. 207, 192–220 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)

    Article  MATH  Google Scholar 

  20. Dowell, E.H., Hall, K.C.: Modeling of fluid-structure interaction. Annu. Rev. Fluid Mech. 33, 445–490 (2001)

    Article  MATH  Google Scholar 

  21. Drmac, Z., Gugercin, S.: A new selection operator for the discrete empirical interpolation method – improved a priori error bound and extensions, SIAM. J. Sci. Comput. 38, A631–A648 (2016)

    MATH  Google Scholar 

  22. Dimitriu, G., Stefanescu, 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34, A937–A969 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Golub, G.H., van Loan, G.T.: Matrix Computations. John Hopkins Univ. Press, Baltimore (1996)

    MATH  Google Scholar 

  25. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)

    Book  MATH  Google Scholar 

  26. Gräßle, C., Hinze, M., Lang, J., Ullmann, S.: POD model order reduction with space-adapted snapshots for incompressible flows. Adv. Comput. Math. 45, 2401–2428 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems. Springer, Berlin (2010)

    MATH  Google Scholar 

  28. Hyman, J.M., Nicolaenko, B.: The Kuramoto-Sivashinsky equation: a bridge between PDE’s and dynamical systems. Physica D 18, 113–126 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kengne, E., Lakhssassi, A., Vaillancourt, R., Liu, W.M.: Exact solutions for generalized variable-coefficients Ginzburg-Landau equation: application to Bose-Einstein condensates with multi-body interatomic interactions, vol. 53 (2012)

  30. Kostova-Vassilevska, T., Oxberry, G.M.: Model reduction of dynamical systems by proper orthogonal decomposition: error bounds and comparison of methods using snapshots from the solution and the time derivatives. J. Comput. Appl. Math. 330, 553–73 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theoret. Phys. 55, 356–369 (1976)

    Article  Google Scholar 

  32. Lam, C.-K., Malomed, B.A., Chow, K.W., Wai, P.K.A.: Spatial solitons supported by localized gain in nonlinear optical waveguides. Eur. Phys. J. Special Topics 173, 233–243 (2009)

    Article  Google Scholar 

  33. Lieu, T., Farhat, C., Lesoinne, M.: Reduced-order fluid/structure modeling of a complete aircraft configuration. Comput. Method. Appl. M. 195, 5730–5742 (2006)

    Article  MATH  Google Scholar 

  34. Malomed, B.A.: Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations, Progress in Optical Science and Photonics. Springer, Berlin (2013)

    MATH  Google Scholar 

  35. Pearson, K.: On lines and planes of closest fit to systems of points in space. Philos. Mag. 2, 559–572 (1901)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Peherstorfer, B., Willcox, K.: Online adaptive model reduction for nonlinear systems via low-rank updates. SIAM J. Sci. Comput. 37, A2123–A2150 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Comput. 10, 777–786 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rapún, M.-L., Terragni, F., Vega, J.M.: Adaptive POD-based low-dimensional modeling supported by residual estimates. Int. J. Numer. Meth. Eng. 9, 844–868 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rapún, M.-L., Terragni, F., Vega, J.M.: LUPOD: Collocation in POD via LU decomposition. J. Comput. Phys. 335, 1–20 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rapún, M.-L., Vega, J.M.: Reduced order models based on local POD plus Galerkin projection. J. Comput. Phys. 229, 3046–3063 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rempfer, D.: On low-dimensional Galerkin models for fluid flow. Theor. Comp. Fluid Dyn. 14, 75–88 (2000)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Sirisup, S., Karniadakis, G.E., Xiu, D., Kevrekidis, I.G.: Equations-free/Galerkin-free POD assisted computation of incompressible flows. J. Comput. Phys. 207, 568–587 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sirovich, L.: Turbulence and the dynamics of coherent structures. Q. Appl. Math. XLV, 561–590 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sivashinsky, G.: Nonlinear analysis of hydrodynamic instability in laminar flames I. Derivation of basic equations. Acta Astron. 4, 1177–1206 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  47. Soll, T., Pulch, R.: Sample selection based on sensitivity analysis in parameterized model order reduction. J. Comput. Appl. Math. 316, 369–379 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stabile, G., Ballarin, F., Zuccarino, G., Rozza, G.: A reduced order variational multiscale approach for turbulent flows. Adv. Comput. Math. 45, 2349–2368 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  49. Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V., Ukeiley, L.S.: Modal analysis of fluid flows: an overview. AIAA J. 55, 4013–4041 (2017)

    Article  Google Scholar 

  50. Terragni, F., Valero, E., Vega, J.M.: Local POD plus Galerkin projection in the unsteady lid-driven cavity problem. SIAM J. Sci. Comput. 33, 3538–3561 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Terragni, F., Vega, J.M.: On the use of POD-based ROMs to analyze bifurcations in some dissipative systems. Physica D 241, 1393–1405 (2012)

    Article  MATH  Google Scholar 

  52. Terragni, F., Vega, J.M.: Construction of bifurcation diagrams using POD on the fly. SIAM J. Appl. Dyn. Syst. 13, 339–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Tobias, S.M., Proctor, M.R.E., Knobloch, E.: Convective and absolute instabilities of fluid flows in finite geometry. Physica D 113, 43–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zimmermann, R.: Towards best-practice guidelines for POD-based reduced order modeling of transonic flows EUROGEN. In: 2011 Proceedings, CIRA (2011)

Download references

Acknowledgements

The authors would like to thank two anonymous referees for some useful comments that helped improve the paper.

Funding

This work has been supported by the FEDER / Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación, under grants TRA2016-75075-R and MTM2017-84446-C2-2-R.

Author information

Authors and Affiliations

  1. E.T.S.I. Aeronáutica y del Espacio, Universidad Politécnica de Madrid, 28040, Madrid, Spain

    M.-L. Rapún & J. M. Vega

  2. G. Millán Institute and Department of Mathematics, Universidad Carlos III de Madrid, 28911, Leganés, Spain

    F. Terragni

Authors
  1. M.-L. Rapún
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Terragni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. M. Vega
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M.-L. Rapún.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by: Stefan Volkwein

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rapún, ML., Terragni, F. & Vega, J.M. Adaptive sampling and modal expansions in pattern-forming systems. Adv Comput Math 47, 48 (2021). https://doi.org/10.1007/s10444-021-09870-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09870-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation