Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T19:29:32.681Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

Published online by Cambridge University Press:  03 June 2015

Zhen Gao  and
Tao Zhou
Zhen Gao*
Affiliation:
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Tao Zhou*
Affiliation:
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
Email:zhengao@ouc.edu.cn
Corresponding author.Email:tzhou@lsec.cc.ac.cn
Get access

Abstract

In this work, we concern with the numerical comparison between different kinds of design points in least square (LS) approach on polynomial spaces. Such a topic is motivated by uncertainty quantification (UQ). Three kinds of design points are considered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points and the Quasi Monte Carlo (QMC) points. We focus on three aspects during the comparison: (i) the convergence properties; (ii) the stability, i.e. the properties of the resulting condition number of the design matrix; (iii) the robustness when numerical noises are present in function values. Several classical high dimensional functions together with a random ODE model are tested. It is shown numerically that (i) neither the MC sampling nor the QMC sampling introduce the low convergence rate, namely, the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used; (ii)The use of SG points admits better convergence properties only for very low dimensional problems (say d ≤ 2); (iii)The QMC points, being deterministic, seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.

Keywords

65C2065M70Least squarepolynomial approximationsuncertainty quantificationcondition number.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 365 - 381
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Blatman, G. and Sudret, B.Sparse polynomial chaos expansions and adaptive stochastic finite elements using regression approach. C.R.Mechanique, 336: 518–523, 2008.CrossRefGoogle Scholar
[2]Chen, Q.-Y., Gottlieb, D. and Hesthaven, J., Uncertainty analysis for the steady-state flows in a dual throat nozzle, J. Comput. Phys., 204 (2005), pp. 387–398.CrossRefGoogle Scholar
[3]Cheng, H. and Sandu, A., Collocation least-squares polynomial chaos method. In Proceedings of the 2010 Spring Simulation Multiconference, pages 94–99, April, 2010.Google Scholar
[4]Cohen, A., Davenport, M.A., and Leviatan, D., On the stability and accuracy of Least Squares approximations, Found. Comput. Math., (2013) 13:819–834.Google Scholar
[5]Doostan, A. and Iaccarino, G., A least-squares approximation of partial differential equations with high-dimensional random inputs, J. Comput. Phys., 228 (2009) 4332–4345.Google Scholar
[6]Eldred, M. S., Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design, Proceedings of the 11th AIAA Nondeterministic Approaches Conference, No. AIAA-2009-2274, Palm Springs, CA, May 4-7 2009.CrossRefGoogle Scholar
[7]Fishman, G., Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996.Google Scholar
[8]Gao, Zhen and Hesthaven, Jan S., Efficient solution of ordinary differential equations with high-dimensional parametrized uncertainty, Commun. Comput. Phys. 10, 253–278,2011.CrossRefGoogle Scholar
[9]Gardner, T., Cantor, C. and Collins, J., Construction of a Genetic Toggle Switch in Escherichia coli, Nature 403,339–342,2000.Google Scholar
[10]Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.Google Scholar
[11]Hammersley, John, Monte Carlo methods for solving multivariable problems, Proceedings of the New York Academy of Science, Volume 86,1960, pages 844–874.CrossRefGoogle Scholar
[12]Hosder, S., Walters, R. W., and Balch, M., Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables, Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, No. AIAA-2007-1939, Honolulu, HI, April 23-26,2007.Google Scholar
[13]Hosder, S., Walters, R. W., and Balch, M.Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA Journal, 48: 2721–2730, 2010.Google Scholar
[14]Kocis, Ladislav and Whiten, William, Computational Investigations of Low-Discrepancy Sequences, ACM Transactions on Mathematical Software, Volume 23, Number 2, 1997, pages 266–294.Google Scholar
[15]Migliorati, G., Nobile, F., Schwerin, E., and Tempone, R., Analysis of the discrete L2 projection on polynomial spaces with random evaluations, Foundations of Computational Mathematics, DOI:10.1007/s10208-013-9186-4, 2014.Google Scholar
[16]Morokoff, W. J. and Caflisch, R. E., Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput., 15(6): 1251–1279, November 1994.Google Scholar
[17]Nobile, F., Tempone, R. and Webster, C., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 2008, vol. 46/5, pp. 2309–2345.Google Scholar
[18]Nobile, F., Tempone, R. and Webster, C., An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 2008, vol. 46/5, pp. 2411–2442.Google Scholar
[19]Novak, E. and Ritter, K., High dimensional integration of smooth functions over cubes, Numer. Math., 75 (1996), pp. 79–97.CrossRefGoogle Scholar
[20]Novak, E. and Ritter, K., Simple cubature formulas with high polynomial exactness, Constructive Approx., 15 (1999), pp. 499–522.Google Scholar
[21]Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963), pp. 240–243.Google Scholar
[22]Wiener, N., The homogeneous chaos, Am. J. Math., 60 (1938), 897–936.CrossRefGoogle Scholar
[23]Xiu, D., Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), 293–309.Google Scholar
[24]Xiu, D. and Hesthaven, J.S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139.Google Scholar
[25]Xiu, D. and Karniadakis, G.E., The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., Vol. 24, No. 2, pp. 619–644.Google Scholar
[26]Xu, Z. and Zhou, T., On sparse interpolation and the design of deterministic interpolation points, arxiv:1308.6038, Aug. 2013.Google Scholar
[27]Zhou, T., Narayan, A. and Xu, Z., Multivariate discrete least-squares approximations with a new type of collocation grid, Available online: http://arxiv.org/abs/1401.0894, Jan. 2014.Google Scholar
[28]Zhou, T. and Tang, T., Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials, J. Sci. Comput., (2012)51:274–292.CrossRefGoogle Scholar