Global sensitivity analysis by polynomial dimensional decomposition

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.

[1]  I. Sobol,et al.  About the use of rank transformation in sensitivity analysis of model output , 1995 .

[2]  Amparo Alonso-Betanzos,et al.  Functional Network Topology Learning and Sensitivity Analysis Based on ANOVA Decomposition , 2007, Neural Computation.

[3]  Jon C. Helton,et al.  Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems , 2002 .

[4]  Shuangzhe Liu,et al.  Global Sensitivity Analysis: The Primer by Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, Stefano Tarantola , 2008 .

[5]  H. Rabitz,et al.  Practical Approaches To Construct RS-HDMR Component Functions , 2002 .

[6]  Olivier P. Le Maître,et al.  Polynomial chaos expansion for sensitivity analysis , 2009, Reliab. Eng. Syst. Saf..

[7]  S. Rahman A polynomial dimensional decomposition for stochastic computing , 2008 .

[8]  Stefano Tarantola,et al.  Random balance designs for the estimation of first order global sensitivity indices , 2006, Reliab. Eng. Syst. Saf..

[9]  A. Saltelli,et al.  Making best use of model evaluations to compute sensitivity indices , 2002 .

[10]  H. Rabitz,et al.  General foundations of high‐dimensional model representations , 1999 .

[11]  A. Saltelli,et al.  Reliability Engineering and System Safety , 2008 .

[12]  Saltelli Andrea,et al.  Sensitivity Analysis for Nonlinear Mathematical Models. Numerical ExperienceSensitivity Analysis for Nonlinear Mathematical Models. Numerical Experience , 1995 .

[13]  Stefano Tarantola,et al.  Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models , 2004 .

[14]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[15]  K. Shuler,et al.  Nonlinear sensitivity analysis of multiparameter model systems , 1977 .

[16]  Paola Annoni,et al.  Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index , 2010, Comput. Phys. Commun..

[17]  Sharif Rahman Statistical Moments of Polynomial Dimensional Decomposition , 2010 .

[18]  George N. Karystinos,et al.  ORTHOGONAL POLYNOMIAL EXPANSIONS FOR SOLVING RANDOM EIGENVALUE PROBLEMS , 2011 .

[19]  W. Hoeffding A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[20]  S. Rahman,et al.  Decomposition methods for structural reliability analysis , 2005 .

[21]  Peter C. Young,et al.  State Dependent Parameter metamodelling and sensitivity analysis , 2007, Comput. Phys. Commun..

[22]  Ilya M. Sobol,et al.  Theorems and examples on high dimensional model representation , 2003, Reliab. Eng. Syst. Saf..

[23]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[24]  S. Rahman,et al.  A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics , 2004 .

[25]  I. Sobol On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[26]  T. Ishigami,et al.  An importance quantification technique in uncertainty analysis for computer models , 1990, [1990] Proceedings. First International Symposium on Uncertainty Modeling and Analysis.

[27]  S. Rahman,et al.  A generalized dimension‐reduction method for multidimensional integration in stochastic mechanics , 2004 .

[28]  Sharif Rahman,et al.  Decomposition methods for structural reliability analysis revisited , 2011 .

[29]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[30]  Saltelli Andrea,et al.  Global Sensitivity Analysis: The Primer , 2008 .

[31]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[32]  Stefano Tarantola,et al.  Calculating first-order sensitivity measures: A benchmark of some recent methodologies , 2009, Reliab. Eng. Syst. Saf..

[33]  D. Hunter Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 2000 .

[34]  Sharif Rahman,et al.  ORTHOGONAL POLYNOMIAL EXPANSIONS FOR SOLVING RANDOM EIGENVALUE PROBLEMS , 2011 .

[35]  A. O'Hagan,et al.  Probabilistic sensitivity analysis of complex models: a Bayesian approach , 2004 .

[36]  F. E. Satterthwaite Random Balance Experimentation , 1959 .

[37]  Sharif Rahman Extended Polynomial Dimensional Decomposition for Arbitrary Probability Distributions , 2009 .