Skip to main content
SpringerLink
Log in

Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion

  • Conference paper
  • First Online:
Sparse Grids and Applications - Stuttgart 2014

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 109))

Abstract

We present new sparse-grid based algorithms for fast Bayesian estimation and inversion of parametric operator equations. We propose Reduced Basis (RB) acceleration of numerical integration based on Smolyak sparse grid quadrature. To tackle the curse-of-dimensionality in high-dimensional Bayesian inversion, we exploit sparsity of the parametric forward solution map as well as of the Bayesian posterior density with respect to the random parameters. We employ an dimension adaptive Sparse Grid method (aSG) for both, offline-training the reduced basis as well as for deterministic quadrature of the conditional expectations which arise in Bayesian estimates. For the forward problem with nonaffine dependence on the random variables, we perform further affine approximation based on the Empirical Interpolation Method (EIM) proposed in [1]. A novel combined algorithm to adaptively refine the sparse grid used for quadrature approximation of the Bayesian estimates, of the reduced basis approximation and to compress the parametric forward solutions by empirical interpolation is proposed. The theoretically predicted computational efficiency which is independent of the number of active parameters is demonstrated in numerical experiments for a model, nonaffine-parametric, stationary, elliptic diffusion problem, in two spacial and in parameter space dimensions up to 1024.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A subset \(\Lambda \subset \mathcal{F}\) is a dc set if for every \(\boldsymbol{\nu }\in \Lambda _{M}\) also \(\boldsymbol{\mu }\in \Lambda _{M}\) for any \(\boldsymbol{\mu }\preceq \boldsymbol{\nu }\) (μ j ≤ν j for all \(j \in \mathbb{J}\))

References

  1. M. Barrault, Y. Maday, N. Nguyen, A. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, Analyse Numérique 339(9), 667–672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457–1472 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Chen, A. Quarteroni, Accurate and efficient evaluation of failure probability for partial differential equations with random input data. Comput. Methods Appl. Mech. Eng. 267(0), 233–260 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Chen, A. Quarteroni, Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraints. SIAM/ASA J. Uncertain. Quantif. 2(1), 364–396 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Chen, A. Quarteroni, A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods. J. Comput. Phys. 298, 176–193 (2015)

    Article  MathSciNet  Google Scholar 

  6. P. Chen, A. Quarteroni, G. Rozza, Comparison of reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59, 187–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Chen, A. Quarteroni, G. Rozza, A weighted empirical interpolation method: a priori convergence analysis and applications. ESAIM: Math. Model. Numer. Anal. 48, 943–953, 7 (2014)

    Google Scholar 

  8. P. Chen, A. Quarteroni, G. Rozza, Reduced order methods for uncertainty quantification problems. ETH Zurich, SAM Report 03, Submitted, 2015

    Google Scholar 

  9. P. Chen, C. Schwab, Sparse grid, reduced basis Bayesian inversion. Comput. Methods Appl. Mech. Eng. 297, 84–115 (2015)

    Article  MathSciNet  Google Scholar 

  10. P. Chen, C. Schwab, Sparse grid, reduced basis Bayesian inversion: nonaffine-parametric nonlinear equations. ETH Zurich, SAM Report 21, Submitted, 2015

    Google Scholar 

  11. A. Chkifa, A. Cohen, R. DeVore, C. Schwab, Adaptive algorithms for sparse polynomial approximation of parametric and stochastic elliptic PDEs. M2AN Math. Mod. Num. Anal. 47(1), 253–280 (2013)

    Google Scholar 

  12. A. Chkifa, A. Cohen, C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric pdes. Journal de Mathématiques Pures et Appliquées. 103(2), 400–428 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Cohen, R. DeVore, Kolmogorov widths under holomorphic mappings (2014). arXiv:1502.06795

    Google Scholar 

  14. A. Cohen, R. DeVore, C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. 9(01), 11–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Cui, Y. Marzouk, K. Willcox, Data-driven model reduction for the bayesian solution of inverse problems. Int. J. Numer. Methods Eng. 102(5), 966–990 (2015)

    Article  MathSciNet  Google Scholar 

  16. M. Dashti, A. Stuart, The Bayesian approach to inverse problems (2016). arXiv:1302.6989, to appear in Springer Handbook of Uncertainty Quantification, Editor: Ghanem et al.

    Google Scholar 

  17. D. Galbally, K. Fidkowski, K. Willcox, O. Ghattas, 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 

  18. T. Gerstner, M. Griebel, Dimension–adaptive tensor–product quadrature. Computing 71(1), 65–87 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Grepl, Y. Maday, N. Nguyen, A. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math. Model. Numer. Anal. 41(03), 575–605 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Hansen, C. Schwab, Sparse adaptive approximation of high dimensional parametric initial value problems. Vietnam J. Math. 41(2), 181–215 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. S. Hesthaven, B. Stamm, S. Zhang, Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM Math. Model. Numer. Anal. 48(1), 259–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Hoang, C. Schwab, Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale pdes. Anal. Appl. (Singap.) 11(1), 1350001 (2013)

    Google Scholar 

  23. V. Hoang, C. Schwab, Sparse tensor galerkin discretizations for parametric and random parabolic pdes – analytic regularity and gpc approximation. SIAM J. Math. Anal. 45(5), 3050–3083 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. V. Hoang, C. Schwab, A. Stuart, Complexity analysis of accelerated mcmc methods for bayesian inversion. Inverse Probl. 29(8), 085010 (2013)

    Google Scholar 

  25. Y. Maday, N. Nguyen, A. Patera, G. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8(1), 383–404 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Y. Maday, A. Patera, D. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems. Stud. Math. Appl. 31, 533–569 (2002)

    MathSciNet  MATH  Google Scholar 

  27. N. Nguyen, G. Rozza, D. Huynh, A. Patera, Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; application to real-time Bayesian parameter estimation. Biegler, Biros, Ghattas, Heinkenschloss, Keyes, Mallick, Tenorio, van Bloemen Waanders, and Willcox, editors, Computational Methods for Large Scale Inverse Problems and Uncertainty Quantification (John Wiley, Hoboken, 2009)

    Google Scholar 

  28. F. Nobile, L. Tamellini, R. Tempone, Convergence of quasi-optimal sparse grid approximation of Hilbert-valued functions: application to random elliptic PDEs. EPFL MATHICSE report 12, 2014

    Google Scholar 

  29. F. Nobile, R. Tempone, C. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. G. Rozza, D. Huynh, A. Patera, 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. C. Schillings, C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems. Inverse Probl. 29(6), 065011 (2013)

    Google Scholar 

  32. C. Schillings, C. Schwab, Sparsity in Bayesian inversion of parametric operator equations. Inverse Probl. 30(6), 065007 (2014)

    Google Scholar 

  33. C. Schwab, A. Stuart, Sparse deterministic approximation of bayesian inverse problems. Inverse Probl. 28(4), 045003 (2012)

    Google Scholar 

  34. A. Stuart, Inverse problems: a Bayesian perspective. Acta Numer. 19(1), 451–559 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. G. Turinici, C. Prud’Homme, A. Patera, Y. Maday, A. Buffa, A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Math. Model. Numer. Anal. 46(3):595 (2012)

    Google Scholar 

  36. D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach (Princeton University Press, Princeton, 2010)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported in part by the European Research Council (ERC) under AdG427277.

Author information

Authors and Affiliations

  1. HG G 56.1, Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092, Zürich, Switzerland

    Peng Chen & Christoph Schwab

Authors
  1. Peng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Schwab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Chen .

Editor information

Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Jochen Garcke

  2. Institute for Parallel and Distributed S, Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Chen, P., Schwab, C. (2016). Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion. In: Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Stuttgart 2014. Lecture Notes in Computational Science and Engineering, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-319-28262-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation