Variance-based global sensitivity analysis via sparse-grid interpolation and cubature

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional (random) parame- ter space. In addition to providing highly accurate stochastic solutions, the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters. In this paper, we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol'. This method al- lows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids. We discuss convergence of this method, apply it to several test cases and compare to existing methods. As a result which may be of independent interest, we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema. This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

[1]  E. E. Myshetskaya,et al.  Monte Carlo estimators for small sensitivity indices , 2008, Monte Carlo Methods Appl..

[2]  J. Dicapua Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.

[3]  Ronald Cools,et al.  Quasi-random integration in high dimensions , 2007, Math. Comput. Simul..

[4]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[5]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[6]  Marco Ratto,et al.  Global Sensitivity Analysis , 2008 .

[7]  T. J. Rivlin The Chebyshev polynomials , 1974 .

[8]  Ilya M. Sobol,et al.  On Global Sensitivity Indices: Monte Carlo Estimates Affected by Random Errors , 2007, Monte Carlo Methods Appl..

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

[10]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[11]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[12]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

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

[14]  A. Saltelli,et al.  A quantitative model-independent method for global sensitivity analysis of model output , 1999 .

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

[16]  Roger M. Cooke,et al.  Sample-based estimation of correlation ratio with polynomial approximation , 2007, TOMC.

[17]  Harvey M. Wagner,et al.  Global Sensitivity Analysis , 1995, Oper. Res..

[18]  D. Xiu Fast numerical methods for stochastic computations: A review , 2009 .