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

  • Zhen Gao (a1) (a2) and Jan S. Hesthaven (a2)

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.

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Corresponding author

Corresponding author. Email: Jan.Hesthaven@Brown.edu

References

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[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.
[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.
[3] Clenshaw, C. W., and Curtis, A. R., A method for numerical integration on an automatic computer, Numer. Math., 2 (1960), 197.
[4] Davis, P. J., and Rabinowitz, P., Methods of Numerical Integration, Academic Press, NY, 1975.
[5] Fishman, G., Monte Carlo: Concepts, Algorithms, and Applications, Springer-Verlag, New York, 1996.
[6] Fox, B., Strategies for Quasi-Monte Carlo, Kluwer, Dordrect, The Netherlands, 1999.
[7] Gamerman, D., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Chapman and Hall, London, 1997.
[8] Gardner, T., Cantor, C., and Collins, J., Construction of a genetic toggle switch in Escherichia coli, Nature., 403 (2000), 339–342.
[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.
[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.
[11] Gerstner, T., and Griebel, M., Numerical integration using sparse grid, Numer. Algor., 18 (1998), 209–232.
[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.
[13] Gu, C., Smoothing Spline ANOVA Models, Springer, Berlin, 2002.
[14] Homma, T., and Saltelli, A., Importance measures in global sensitivity analysis of nonlinear models, Reliab. Eng. Syst. Safe., 52 (1996), 1–17.
[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.
[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.
[17] Owen, A. B., The dimension distribution and quadrature test functions, Technical Report, Stanford University, 2001.
[18] Paskov, S., and Traub, J., Faster valuation of financial derivatives, J. Portfolio. Manag., 22 (1995), 113–120.
[19] Trefethen, L. N., Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50 (2008), 67–87.
[20] Saltelli, A., Chan, K., and Scott, E., Sensitivity Analysis, Wiley & Sons, Chichester, 2000.
[21] Smolyak, S. A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk. SSSR., 4 (1963), 240–243.
[22] Sobol, I. M.’, Sensitivity estimates for nonlinear mathematical models, Math. Model., 2 (1990), 112–118 (in Russian).
[23] Sobol’, I. M., Sensitivity estimates for nonlinear mathematical models, Math. Model. Comput. Exp., 1 (1993), 407–414.
[24] Stein, M., Large sample properties of simulations using Latin hypercube sampling, Techno-metrics, 29 (1987), 143–151.
[25] Stroud, A., Remarks on the disposition of points in numerical integration formulas, Math. Comput., 11 (1957), 257–261.
[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.
[27] Verwer, J. G., Gauss-Seidel iteration for stiff ODEs from chemical kinetics, SIAM J. Sci. Comput., 15 (1994), 1243–1259.
[28] Wang, X., and Fang, K., The effective dimension and quasi-Monte Carlo integration, J. Complex., 19 (2003), 101–124.
[29] Wasilakowski, G. W., and Wozniakowski, H., Explicit cost bounds of algorithms for multi-variate tensor product problems, J. Complex., 11 (1995), 1–56.
[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.
[31] Xiu, D., Efficient collocational approachfor parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), 293–309.
[32] Xiu, D., Numerical integration formula of degree two, Appl. Numer. Math., 58 (2008), 1515–1520.
[33] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), 242–272.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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