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Efficient Solution of Ordinary Differential Equations with High-Dimensional Parametrized Uncertainty

Published online by Cambridge University Press:  20 August 2015

Zhen Gao  and
Jan S. Hesthaven
Zhen Gao*
Affiliation:
Research Center for Applied Mathematics, Ocean University of China, Qingdao 266071, Shandong, China Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
Jan S. Hesthaven*
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
*
Email:zhengao.ouc@gmail.com
Corresponding author.Email:Jan.Hesthaven@Brown.edu
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Abstract

The important task of evaluating the impact of random parameters on the output of stochastic ordinary differential equations (SODE) can be computationally very demanding, in particular for problems with a high-dimensional parameter space. In this work we consider this problem in some detail and demonstrate that by combining several techniques one can dramatically reduce the overall cost without impacting the predictive accuracy of the output of interests. We discuss how the combination of ANOVA expansions, different sparse grid techniques, and the total sensitivity index (TSI) as a pre-selective mechanism enables the modeling of problems with hundred of parameters. We demonstrate the accuracy and efficiency of this approach on a number of challenging test cases drawn from engineering and science.

Keywords

62F1265C2065C30Sparse gridsstochastic collocation methodANOVA expansiontotal sensitivity index
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 253 - 278
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Caflisch, R. E., Morokoff, W., and Owen, A., Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance., 1 (1997), 27–46.Google Scholar
[2] Cao, Y., Chen, Z., and Gunzburger, M., ANOVA expansions and efficient sampling methods for parameter dependent nonlinear PDEs, Int. J. Numer. Anal. Model., 6 (2009), 256–273.Google Scholar
[3] Clenshaw, C. W., and Curtis, A. R., A method for numerical integration on an automatic computer, Numer. Math., 2 (1960), 197.Google Scholar
[4] Davis, P. J., and Rabinowitz, P., Methods of Numerical Integration, Academic Press, NY, 1975.Google Scholar
[5] Fishman, G., Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
[6] Fox, B., Strategies for Quasi-Monte Carlo, Kluwer, Dordrect, The Netherlands, 1999.CrossRefGoogle Scholar
[7] Gamerman, D., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Chapman and Hall, London, 1997.Google Scholar
[8] Gardner, T., Cantor, C., and Collins, J., Construction of a genetic toggle switch in Escherichia coli, Nature., 403 (2000), 339–342.Google Scholar
[9] Genz, A. C., Testing multidimensional integration routines, in: Tools, Methods, and Languages for Scientific and Engineering Computation, eds. Ford, B., Rault, J. C., and Thomasset, F. (North-Holland, Amsterdam, 1984), 81–94.Google Scholar
[10] Genz, A. C., A package for testing multiple integration subroutines, in: Numerical Integration, eds. Keast, P., and Fairweather, G. (Kluwer, Dordrecht, 1987), 337–340.Google Scholar
[11] Gerstner, T., and Griebel, M., Numerical integration using sparse grid, Numer. Algor., 18 (1998), 209–232.CrossRefGoogle Scholar
[12] Giles, M. B., and Suli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta. Numer., 11 (2002), 145–206.CrossRefGoogle Scholar
[13] Gu, C., Smoothing Spline ANOVA Models, Springer, Berlin, 2002.Google Scholar
[14] Homma, T., and Saltelli, A., Importance measures in global sensitivity analysis of nonlinear models, Reliab. Eng. Syst. Safe., 52 (1996), 1–17.CrossRefGoogle Scholar
[15] Liu, M., Gao, Z., and Hesthaven, J. S., Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty, Appl. Numer. Math., 2010, submitted.Google Scholar
[16] Mazzia, F., and Magherini, C., Test Set for Initial Values Problem Solvers, Release 2.4, Department of Mathematics, University of Bari and INdAM, Research Unit of Bari, February 2008.Google Scholar
[17] Owen, A. B., The dimension distribution and quadrature test functions, Technical Report, Stanford University, 2001.Google Scholar
[18] Paskov, S., and Traub, J., Faster valuation of financial derivatives, J. Portfolio. Manag., 22 (1995), 113–120.CrossRefGoogle Scholar
[19] Trefethen, L. N., Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50 (2008), 67–87.CrossRefGoogle Scholar
[20] Saltelli, A., Chan, K., and Scott, E., Sensitivity Analysis, Wiley & Sons, Chichester, 2000.Google Scholar
[21] Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk. SSSR., 4 (1963), 240–243.Google Scholar
[22] Sobol, I. M.’, Sensitivity estimates for nonlinear mathematical models, Math. Model., 2 (1990), 112–118 (in Russian).Google Scholar
[23] Sobol’, I. M., Sensitivity estimates for nonlinear mathematical models, Math. Model. Comput. Exp., 1 (1993), 407–414.Google Scholar
[24] Stein, M., Large sample properties of simulations using Latin hypercube sampling, Techno-metrics, 29 (1987), 143–151.Google Scholar
[25] Stroud, A., Remarks on the disposition of points in numerical integration formulas, Math. Comput., 11 (1957), 257–261.Google Scholar
[26] Tatang, M. A., Pan, W. W., Prinn, R. G., and McRae, G. J., An efficient method for parametric uncertainty analysis of numerical geophysical model, J. Geophy. Res., 102 (1997), 21925–21932.CrossRefGoogle Scholar
[27] Verwer, J. G., Gauss-Seidel iteration for stiff ODEs from chemical kinetics, SIAM J. Sci. Comput., 15 (1994), 1243–1259.CrossRefGoogle Scholar
[28] Wang, X., and Fang, K., The effective dimension and quasi-Monte Carlo integration, J. Complex., 19 (2003), 101–124.Google Scholar
[29] Wasilakowski, G. W., and Wozniakowski, H., Explicit cost bounds of algorithms for multi-variate tensor product problems, J. Complex., 11 (1995), 1–56.Google Scholar
[30] Xiu, D., and Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118–1139.Google Scholar
[31] Xiu, D., Efficient collocational approachfor parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), 293–309.Google Scholar
[32] Xiu, D., Numerical integration formula of degree two, Appl. Numer. Math., 58 (2008), 1515–1520.Google Scholar
[33] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), 242–272.Google Scholar