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Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature

Published online by Cambridge University Press:  20 August 2015

Gregery T. Buzzard  and
Dongbin Xiu
Gregery T. Buzzard*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Dongbin Xiu*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional (random) parameter space. In addition to providing highly accurate stochastic solutions, the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters. In this paper, we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’. This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids. We discuss convergence of this method, apply it to several test cases and compare to existing methods. As a result which may be of independent interest, we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema. This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

Keywords

41A1041A0541A6341A55Stochastic collocationsparse gridssensitivity analysisSmolyakSobol’
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 542 - 567
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Barthelmann, V., Novak, E. and Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comp. Math., 12 (2000), 273–288.Google Scholar
[2]Crestaux, T., Le, O. Matre and Martinez, J. M., Polynomial chaos expansion for sensitivity analysis, Reliab. Eng. Syst. Safe., 94 (2009), 1161–1172.CrossRefGoogle Scholar
[3]Gerstner, T. and Griebel, M., Numerical integration using sparse grids, Numer. Algorithms., 18 (1998), 209–232.Google Scholar
[4]Lewandowski, D., Cooke, R. M. and Tebbens, R. J. D., Sample-based estimation of correlation ratio with polynomial approximation, ACM Trans. Model. Comp. Simul., 18 (2007), 1–17.Google Scholar
[5]Mason, J. C. and Handscomb, D. C., Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, 2003.Google Scholar
[6]Saltelli, A., Tarantola, S. and Chan, K. P. S., A quantitative model-independent method for global sensitivity analysis of model output, Technometrics., 41 (1999), 39–54.CrossRefGoogle Scholar
[7]Saltelli, A., Making best use of model evaluations to compute sensitivity indices, Comp. Phys. Comm., 145 (2002), 280–297.Google Scholar
[8]Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M. and Tarantola, S., Global Sensitivity Analysis, The Primer, John Wiley & Sons, Ltd, 2008.Google Scholar
[9] Simlab 3.2.6, http://simlab.jrc.ec.europa.eu/, accessed 7/21/2009.Google Scholar
[10]Sobol´, I. M., Sensitivity estimates for nonlinear mathematical models, Math. Model. Comp. Exp., 1 (1993), 407–414, [Translation of Sensitivity estimates for nonlinear mathematical models, Matematicheskoe Modelirovanie, 2 (1990), 112–118 (in Russian)].Google Scholar
[11]Sobol´, I. M. and Myshetskaya, E. E., Monte Carlo estimators for small sensitivity indices, Monte. Carlo. Methods. Appl., 13 (2007), 455–465.Google Scholar
[12]Takhtamysheva, G., Vandewoestyneb, B. and Cools, R., Quasi-random integration in high dimensions, Math. Comput. Simulat., 73 (2007), 309–319.CrossRefGoogle Scholar
[13]Trefethen, L. N., Is Gauss quadrature better than Clensh-Curtis?, SIAM Rev., 50 (2008), 67–87.Google Scholar
[14]Weisstein, E. W., Christoffel-Darboux Identity, MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/Christoffel-DarbouxIdentity.html.Google Scholar
[15]Xiu, D., Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys., 5 (2009), 242–272.Google Scholar
[16]Xiu, D. and Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118–1139.CrossRefGoogle Scholar