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 Probability Surveys > Vol. 10 (2013) open journal systems 


On spectral methods for variance based sensitivity analysis

Alen Alexanderian, The University of Texas at Austin


Abstract
Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.

Keywords: Variance based sensitivity analysis, analysis of variance, spectral methods, generalized polynomial chaos, orthogonal polynomials, conditional expectation.

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Alexanderian, Alen, On spectral methods for variance based sensitivity analysis, Probability Surveys, 10, (2013), 51-68 (electronic). DOI: 10.1214/13-PS219.

References

[1]    Abramowitz, M. and Stegun, I.A.. Handbook of Mathematical Functions. Dover, New York, 9th edition, 1972.

[2]    Alexanderian, A., Le Maître, O.P., Najm, H.N., Iskandarani, M., and Knio, O.M. Multiscale stochastic preconditioners in non-intrusive spectral projection. Journal of Scientific Computing, 50:306–340, 2012. MR2886330

[3]    Alexanderian, A., Rizzi, F., Rathinam, M., Le Maître, O., and Knio, O. Preconditioned Bayesian regression for stochastic chemical kinetics. Journal of Scientific Computing, 2013. In press.

[4]    Alexanderian, A., Winokur, J., Sraj, I., Srinivasan, A., Iskandarani, M., Thacker,  W.C., and Knio, O.M. Global sensitivity analysis in ocean global circulation models: A sparse spectral projection approach. Computational Geosciences, 16(3):757–778, 2012.

[5]    Archer, G.E.B., Saltelli, A., and Sobol, I.M. Sensitivity measures, ANOVA-like techniques and the use of bootstrap. Journal of Statistical Computation and Simulation, 58(2):99–120, 1997.

[6]    Athreya, K.B. and Lahiri, S.N. Measure Theory and Probability Theory. New York, NY: Springer, 2006. MR2247694

[7]    Cameron, R.H. and Martin, W.T. The orthogonal development of non-linear functionals in series of fourier-hermite functionals. Ann. Math., 48:385–392, 1947. MR0020230

[8]     Crestaux, T., Le Maitre, O.P., and Martinez, J.-M. Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 94(7):1161 – 1172, 2009. Special Issue on Sensitivity Analysis.

[9]    Ernst, O.G., Mugler, A., Starkloff, H.-J. and Ullmann, E. On the convergence of generalized polynomial chaos expansions. ESAIM: Mathematical Modelling and Numerical Analysis, 46:317–339, 2012. MR2855645

[10]    Ghanem, R.G. and Spanos, P.D. Stochastic Finite Elements: A Spectral Approach. Dover, 2nd edition, 2002.

[11]    Homma, T. and Saltelli, A. Importance measures in global sensitivity analysis of nonlinear models. Reliability Engineering & System Safety, 52(1):1–17, 1996.

[12]    Janson, S. Gaussian Hilbert Spaces. Cambridge University Press, 1997. MR1474726

[13]    Kallenberg, O. Foundations of Modern Probability. Springer, 2nd edition, 2002. MR1876169

[14]    Le Maître, O.P. and Knio, O.M. Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics. Scientific Computation, Springer, 2010. MR2605529

[15]    Le Maître, O.P., Mathelin, L., Knio, O.M., and Hussaini, M.Y. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete and Continuous Dynamical Systems, 28(1):199–226, 2010. MR2629479

[16]    Le Maître, O.P., Najm, H.N., Pebay, P.P., Ghanem, R.G., and Knio, O.M. Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM Journal on Scientific Computing, 29(2):864–889, 2007. MR2306272

[17]    Najm, H.N. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech., 41:35–52, 2009. MR2512381

[18]    Reagan, M.T., Najm, H.N., Debusschere, B.J., Le Maître, O.P. Knio, O.M., and Ghanem, R.G. Spectral stochastic uncertainty quantification in chemical systems. Combustion Theory and Modelling, 8:607–632, 2004.

[19]    Saltelli, A. Sensitivity analysis for importance assessment. Risk Analysis, 22(3):579–590, 2002.

[20]    Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S. Global Sensitivity Analysis: The Primer. Wiley.com, 2008. MR2382923

[21]    Sargsyan, K., Debusschere, B., Najm, H.N., and Le Maître, O.P. Spectral representation and reduced order modeling of the dynamics of stochastic reaction networks via adaptive data partitioning. SIAM J. Sci. Comput., 31:4395–4421, 2010. MR2594987

[22]    Sobol, I.M. Estimation of the sensitivity of nonlinear mathematical models. Matematicheskoe Modelirovanie, 2(1):112–118, 1990. MR1052836

[23]    Sobol, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55(1–3):271–280, 2001. The Second IMACS Seminar on Monte Carlo Methods. MR1823119

[24]    Sobol, I.M., Tarantola, S., Gatelli, D., Kucherenko, S.S., and Mauntz, W. Estimating the approximation error when fixing unessential factors in global sensitivity analysis. Reliability Engineering & System Safety, 92(7):957–960, 2007.

[25]    Sudret, B. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 93(7):964–979, 2008.

[26]    Wiener, N. The Homogeneous chaos. Amer. J. Math., 60:897–936, 1938. MR1507356

[27]    Williams, D. Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. MR1155402

[28]    Xiu, D.B. and Karniadakis, G.E. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24:619–644, 2002. MR1951058




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