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Distributional Sensitivity for Uncertainty Quantification

Published online by Cambridge University Press:  20 August 2015

Akil Narayan  and
Dongbin Xiu
Akil Narayan*
Affiliation:
Department of Mathematics, Purdue University, 150, N. University Street, West Lafayette, IN 47904, USA
Dongbin Xiu*
Affiliation:
Department of Mathematics, Purdue University, 150, N. University Street, West Lafayette, IN 47904, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numericalmodel leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

Keywords

52B1065D1868U0568U07Uncertainty quantificationepistemic uncertaintydistributional sensitivitygeneralized polynomial chaos
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 140 - 160
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Arsham, H., Feuerverger, A., McLeish, D. L., Kreimer, J., and Rubinstein, R. Y., Sensitivity analysis and the “what if” problem in simulation analysis, Math. Comput. Model., 12(2) (1989), 193–219.Google Scholar
[2]Asmussen, S., and Glynn, Peter W., Stochastic Simulation: Algorithms and Analysis, Springer, 2007.CrossRefGoogle Scholar
[3]Berger, J., Discussion: on the consistency of bayes estimates, Annal. tat., 14(1) (1986), 30–37.Google Scholar
[4]Berger, J. O., Robust bayesian analysis: sensitivity to the prior, J. tat. Plan. Infer., 25(3) (1990), 303–328.Google Scholar
[5]Diaconis, P., and D. reedman, On the consistency of bayes estimates, Annal. Stat., 14(1) (1986), 1–26.Google Scholar
[6]Ghanem, Roger G., and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, Inc., 1991.Google Scholar
[7]Gibbs, A. L., and Edward Su, F., On choosing and bounding probability metrics, Int. Stat. Rev., 70(3) (2002), 419–435.Google Scholar
[8]Glynn, P. W., Likelihood ratio derviative estimators for stochastic systems, in Proceedings of the 21st Conference on Winter Simulation, pages 374–380, Washington D. C. United States, 1989, ACM.Google Scholar
[9]Gustafson, P., Local sensitivity of posterior expectations, Annal. tat., 24(1) (1996), 174–195.Google Scholar
[10]Marzouk, Y., and Dongbin Xiu, A stochastic collocation approach to bayesian inference in inverse problems, Commun. Comput. Phys., 6(4) (2009), 826–847.Google Scholar
[11]Rachev, S. T., and Romisch, W., Quantitative stability in stochastic programming: the method of probability metrics, Math. Operat. Res., 27(4) (2002), 792–818.Google Scholar
[12]Romisch, W., and Schultz, R., Lipschitz stability for stochastic programs with complete recourse, SIAM J. Opt., 6(2) (1996), 531–547.Google Scholar
[13]Rubinstein, R. Y., Sensitivity analysis and performance extrapolation for computer simulation models, Operat. es., 37(1) (1989), 72–81.Google Scholar
[14]Ruggeri, F., and Wasserman, L., Infinitesimal sensitivity of posterior distributions, Canadian J. Stat., 21(2) (1993), 195–203.Google Scholar
[15]Xiu, Dongbin, Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5(2-4) (2009), 242–272.Google Scholar
[16]Xiu, Dongbin, and Em Karniadakis, G., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24(2) (2002), 619–644.Google Scholar
[17]Xiu, Dongbin, and Em Karniadakis, G., Supersensitivity due to uncertain boundary conditions, Int. J. Numer. Methods. Eng., 61(12) (2004), 2114–2138.Google Scholar
[18]Xiu, Dongbin, and Jie Shen, Efficient stochastic galerkin methods for random diffusion equations, J. Comput. Phys., 228(2) (2009), 266–281.CrossRefGoogle Scholar
[19]Zazanis, M. A., and Suri, R., Convergence rates of finite-difference sensitivity estimates for stochastic systems, Operat. Res., 41(4) (1993), 694–703.Google Scholar