Skip to main content
SpringerLink
Log in

A probabilistic reduced basis method for parameter-dependent problems

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

Abstract

Probabilistic variants of model order reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic reduced basis method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation of some parameter-dependent random variable. Practical algorithms relying on Monte Carlo estimates of this error indicator are discussed. In particular, when using probably approximately correct (PAC) bandit algorithm, the resulting procedure is proven to be a weak-greedy algorithm with high probability. Intended applications concern the approximation of a parameter-dependent family of functions for which we only have access to (noisy) pointwise evaluations. As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula.

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. Nouy, A.: Low-rank tensor methods for model order reduction. SpringerLink, pp. 857–882 (2017). https://doi.org/10.1007/978-3-319-12385-1_21

  2. Cohen, A., Dahmen, W., DeVore, R., Nichols, J.: Reduced basis greedy selection using random training sets. ESAIM: Math. Model. Numer. Anal. 54(5), 1509–1524 (2020)

  3. Cai, D., Yao, C., Liao, Q.: A stochastic discrete empirical interpolation approach for parameterized systems. Symmetry 14(3), 556 (2022). https://doi.org/10.3390/sym14030556

    Article  Google Scholar 

  4. Boyaval, S., Lelièvre, T.: A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm. Commun. Math. Sci. 8(3), 735–762 (2010). https://doi.org/10.4310/CMS.2010.v8.n3.a7

    Article  MathSciNet  Google Scholar 

  5. Blel, M.-R., Ehrlacher, V., Lelièvre, T.: Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables. (2021). arXiv:2105.14091

  6. Homescu, C., Petzold, L.R., Serban, R.: Error estimation for reduced-order models of dynamical systems. SIAM Rev. 49(2), 277–299 (2007)

    Article  MathSciNet  Google Scholar 

  7. Janon, A., Nodet, M., Prieur, C., Prieur, C.: Goal-oriented error estimation for parameter-dependent nonlinear problems. ESAIM: Math. Model. Numer. Anal. 52(2), 705–728 (2018)

  8. Smetana, K., Zahm, O.: Randomized residual-based error estimators for the proper generalized decomposition approximation of parametrized problems. Int. J. Numer. Methods Eng. 121(23), 5153–5177 (2020). https://doi.org/10.1002/nme.6339

    Article  MathSciNet  Google Scholar 

  9. Smetana, K., Zahm, O., Patera, A.T.: Randomized residual-based error estimators for parametrized equations. SIAM J. Sci. Comput. 41(2), 900–926 (2019)

    Article  MathSciNet  Google Scholar 

  10. Balabanov, O., Nouy, A.: Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation. Adv. Comput. Math. 45(5-6), 2969–3019 (2019). https://doi.org/10.1007/s10444-019-09725-6

  11. Balabanov, O., Nouy, A.: Randomized linear algebra for model reduction. Part II: minimal residual methods and dictionary-based approximation. Adv. Comput. Math. 47(2), 26–54 (2021). https://doi.org/10.1007/s10444-020-09836-5

  12. Balabanov, O., Nouy, A.: Preconditioners for model order reduction by interpolation and random sketching of operators. arXiv (2021). https://doi.org/10.48550/ARXIV.2104.12177. https://arxiv.org/abs/2104.12177

  13. Saibaba, A.K.: Randomized discrete empirical interpolation method for nonlinear model reduction. SIAM J. Sci, Comput (2020)

    Book  Google Scholar 

  14. Zahm, O., Nouy, A.: Interpolation of inverse operators for preconditioning parameter-dependent equations. SIAM J. Scie. Comput. 38(2), 1044–1074 (2016) http://dx.doi.org/10.1137/15M1019210. https://doi.org/10.1137/15M1019210

  15. Billaud-Friess, M., Macherey, A., Nouy, A., Prieur, C.: A PAC algorithm in relative precision for bandit problem with costly sampling. Math. Methods Oper. Res. 96(2), 161–185 (2022)

    Article  MathSciNet  Google Scholar 

  16. Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)

    Article  MathSciNet  Google Scholar 

  17. Tyrtyshnikov, E.: Incomplete cross approximation in the mosaic-skeleton method. Computing 64(4), 367–380 (2000). https://doi.org/10.1007/s006070070031

    Article  MathSciNet  Google Scholar 

  18. Billaud-Friess, M., Macherey, A., Nouy, A., Prieur, C.: Stochastic methods for solving high-dimensional partial differential equations. In: International conference on monte Carlo and quasi-Monte Carlo methods in scientific computing, Springer, pp. 125–141 (2018)

  19. Haasdonk, B.: 2. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Reduced basis methods for parametrized PDEs–a tutorial introduction for stationary and instationary problems. SIAM, Philadelphia, PA (2017)

    Chapter  Google Scholar 

  20. Nouy, A.: 4. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Low-rank methods for high-dimensional approximation and model order reduction. SIAM, Philadelphia, PA (2017)

    Chapter  Google Scholar 

  21. 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 (2011)

    Google Scholar 

  22. Buffa, A., Maday, Y., Patera, A.T., Prud’homme, C., Turinici, G.: A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46(3), 595–603 (2012). https://doi.org/10.1051/m2an/2011056

  23. DeVore, R., Petrova, G., Wojtaszczyk, P.: Greedy algorithms for reduced bases in Banach spaces. Constr. Approx. 37(3), 455–466 (2013). https://doi.org/10.1007/s00365-013-9186-2

    Article  MathSciNet  Google Scholar 

  24. Cohen, A., Dahmen, W., Welper, G.: Adaptivity and variational stabilization for convection-diffusion equations. ESAIM: M2AN 46(5), 1247–1273 (2012). https://doi.org/10.1051/m2an/2012003

  25. Lattimore, T., Szepesvári, C.: Bandit algorithms. Cambridge University Press, Cambridge, England (2020)

    Book  Google Scholar 

  26. Friedman, A.: Stochastic differential equations and applications. In: Stochastic differential equations, pp. 75–148. Springer, Berlin, Germany (2010). https://doi.org/10.1007/978-3-642-11079-5_2

  27. Gobet, E.: Monte-Carlo methods and stochastic processes: from linear to non-linear. CRC Press, Boca Raton, FL, USA (2016)

    Book  Google Scholar 

  28. Gobet, E., Maire, S.: A spectral Monte Carlo method for the Poisson equation. De Gruyter 10(3–4), 275–285 (2004). https://doi.org/10.1515/mcma.2004.10.3-4.275

    Article  MathSciNet  Google Scholar 

  29. Gobet, E., Maire, S.: Sequential control variates for functionals of Markov processes on JSTOR. SIAM J. Numer. Anal. 43(3), 1256–1275 (2006)

    Article  Google Scholar 

  30. Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general multipurpose interpolation procedure: the magic points. CPAA 8(1), 383–404 (2008). https://doi.org/10.3934/cpaa.2009.8.383

    Article  MathSciNet  Google Scholar 

  31. Cohen, A., Migliorati, G.: Optimal weighted least-squares methods. SMAI Journal of Computational Mathematics 3, 181–203 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centrale Nantes, Nantes Université, LMJL UMR CNRS 6629, Nantes, France

    Marie Billaud-Friess, Arthur Macherey & Anthony Nouy

  2. University Grenoble Alpes, CNRS, Inria, Grenoble INP*, LJK, 38000, Grenoble, France

    Arthur Macherey & Clémentine Prieur

Authors
  1. Marie Billaud-Friess
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arthur Macherey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anthony Nouy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Clémentine Prieur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marie Billaud-Friess.

Ethics declarations

Competing of interests

The authors declare no competing interests.

Additional information

Communicated by: Tobias Breiten

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Billaud-Friess, M., Macherey, A., Nouy, A. et al. A probabilistic reduced basis method for parameter-dependent problems. Adv Comput Math 50, 19 (2024). https://doi.org/10.1007/s10444-024-10114-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-024-10114-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation