Skip to main content
SpringerLink
Log in

Analysis of Discrete \(L^2\) Projection on Polynomial Spaces with Random Evaluations

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted \(L^2\) norm of the error committed by the random discrete projection is bounded with high probability from above by the best \(L^\infty \) error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. M. Ainsworth and J.T. Oden. A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2000.

  2. I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev., 52(2):317–355, 2010.

    Google Scholar 

  3. J. Bäck, F. Nobile, L. Tamellini, and R. Tempone. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In J.S. Hesthaven and E.M. Ronquist, editors, Spectral and High Order Methods for Partial Differential Equations, volume 76, pages 43–62. Springer, Berlin, 2011.

  4. A.M. Bagirov, C. Clausen, and M. Kohler. Estimation of a regression function by maxima of minima of linear functions. IEEE Trans. Inform. Theory, 55(2):833–845, 2009.

    Google Scholar 

  5. W. Bangerth and R. Rannacher. Adaptive finite element methods for differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2003.

  6. J. Beck, F. Nobile, L. Tamellini, and R. Tempone. On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Mod. Methods, Appl. Sci. (M3AS), 22(9):1250023–1, 1250023–33, 2012.

    Google Scholar 

  7. P. Binev, A. Cohen, W. Dahmen, and R. DeVore. Universal algorithms for learning theory. II. Piecewise polynomial functions. Constr. Approx., 26(2):127–152, 2007.

    Google Scholar 

  8. P. Binev, A. Cohen, W. Dahmen, R. DeVore, and Vladimir Temlyakov. Universal algorithms for learning theory. I. Piecewise constant functions. J. Mach. Learn. Res., 6:1297–1321, 2005.

    Google Scholar 

  9. G. Blatman and B. Sudret. Sparse Polynomial Chaos expansions and adaptive stochastic finite elements using regression approach. C.R.Mechanique, 336:518–523, 2008.

  10. G. Blatman and B. Sudret. Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics, 230(6):2345–2367, 2011.

    Google Scholar 

  11. E.J. Candès, J.K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59(8):1207–1223, 2006.

    Google Scholar 

  12. C. Canuto and A. Quarteroni. Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp., 38(157):67–86, 1982.

    Google Scholar 

  13. A. Cohen, M. Davenport, and D. Leviatan. On the stability and accuracy of least squares approximations. Found. Comput. Math., 13:819–834, 2013.

    Google Scholar 

  14. A. Cohen, R. DeVore, and C. Schwab. Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math., 10(6):615–646, 2010.

    Google Scholar 

  15. A. Cohen, R. DeVore, and C. Schwab. Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. (Singap.), 9(1):11–47, 2011.

    Google Scholar 

  16. T. Crestaux, O. Le Maître, and J.M. Maritinez. Polynomial Chaos Expansion for Sensitivity Analysis. Reliability Engineering and System Safety, 94(7):1161–1172, 2009.

  17. H. A. David and H. N. Nagaraja. Order statistics. Wiley-Interscience, Hoboken, Third edition, 2003.

  18. B. Debusschere, H. Najm, P. Pebray, O. Knio, R. Ghanem, and O. Le Maître. Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes. SIAM J. Sci. Comput., 26(2):698–719, 2005.

  19. D.L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289–1306, 2006.

    Google Scholar 

  20. M.S. Eldred. Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design. In Proceedings of the AIAA 50th conference on Structural Dynamics, and Materials 2009, Palm Springs, California, American Institute of Aeronautics and Astronautics, Reston, VA, 2009.

  21. M.S. Eldred. Design under uncertainty employing stochastic expansion methods. International Journal for Uncertainty Quantification, 1:119–146, 2011.

    Google Scholar 

  22. O.G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann. On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal., 46:317–339, 2012.

    Google Scholar 

  23. G.R. Ghanem and P.D. Spanos. Stochastic Finite Elements: a spectral approach. Dover Publications, Mineola, 2003.

  24. A. Gut. Probability: A Graduate Course, volume 75 of Springer texts in statistics. Springer, Berlin, 2005.

  25. L. Gyørfi, M. Kohler, A. Krzyżak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. Springer-Verlag, Berlin, first edition, 2002.

  26. S. Hosder, R. W. Walters, and M. Balch. Point-Collocation Nonintrusive Polynomial Chaos Method for Stochastic Computational Fluid Dynamics. AIAA Journal, 48:2721–2730, 2010.

    Google Scholar 

  27. M. Kohler. Inequalities for uniform deviations of averages from expectations with applications to nonparametric regression. J. Statist. Plann. Inference, 89(1–2):1–23, 2000.

    Google Scholar 

  28. M. Kohler. Nonlinear orthogonal series estimates for random design regression. J. Statist. Plann. Inference, 115(2):491–520, 2003.

    Google Scholar 

  29. O.P. Le Maître and O.M. Knio. Spectral Methods for Uncertainty Quantification. Springer, New York, 2010.

  30. G. Migliorati. Polynomial approximation by means of the random discrete \(L^{2}\) projection and application to inverse problems for PDEs with stochastic data. PhD thesis, Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano and Centre de Mathématiques Appliquées, École Polytechnique, 2013.

  31. G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone. Approximation of Quantities of Interest in stochastic PDEs by the random discrete \(L^{2}\) projection on polynomial spaces. SIAM J. Sci. Comput., 35(3):A1440–A1460, 2013.

    Google Scholar 

  32. F. Nobile, R. Tempone, and C.G. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2309–2345, 2008.

    Google Scholar 

  33. W. Rudin. Principles of mathematical analysis. McGraw-Hill Book Co., New York, third edition, 1976.

  34. B. Sudret. Global sensitivity analysis using polynomial chaos expansion. Reliability Engineering and System Safety, 93:964–979, 2008.

    Google Scholar 

  35. D. Xiu. Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys., 5(2–4):242–272, 2009.

    Google Scholar 

  36. D. Xiu and G.E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24(2):619–644 (electronic), 2002.

    Google Scholar 

Download references

Acknowledgments

The authors would like to recognize the support of the PECOS Center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project “Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar” and King Abdullah University of Science and Technology (KAUST) through the AEA projects “Predictability and Uncertainty Quantification for Models of Porous Media” and “Tracking Uncertainties in Computational Modeling of Reactive Systems” is also acknowledged. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification. The first and second authors were supported by the Italian Grant FIRB-IDEAS (Project RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems.” We are indebted to A. Cohen and R. DeVore for giving us valuable feedback on the convergence proof. We would also like to thank the anonymous referees for their useful comments that helped us to improve considerably the manuscript, and in particular for the suggestion in the proof of Theorem 2.

Author information

Authors and Affiliations

  1. CSQI-MATHICSE, École Polytechnique Fédérale de Lausanne, 1015 , Lausanne, Switzerland

    G. Migliorati & F. Nobile

  2. Applied Mathematics and Computational Sciences, and SRI Center for Uncertainty Quantification in Computational Science and Engineering, KAUST, Thuwal, 23955-6900, Saudi Arabia

    E. von Schwerin & R. Tempone

Authors
  1. G. Migliorati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Nobile
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. von Schwerin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Tempone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Migliorati.

Additional information

Communicated by Albert Cohen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Migliorati, G., Nobile, F., von Schwerin, E. et al. Analysis of Discrete \(L^2\) Projection on Polynomial Spaces with Random Evaluations. Found Comput Math 14, 419–456 (2014). https://doi.org/10.1007/s10208-013-9186-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-013-9186-4

Keywords

Mathematical Subject Classification

Search

Navigation