Computationally efficient uncertainty propagation and reduction using the stochastic response surface method

Abstract

A computationally efficient means for propagation of uncertainty in computational models is provided by the stochastic response surface method (SRSM), which facilitates uncertainty analysis through the determination of statistically equivalent reduced models. SRSM expresses random outputs in terms of a "polynomial chaos expansion" of Hermite polynomials, and uses an efficient collocation scheme with regression to determine the coefficients of the expansion. This polynomial form then allows straightforward determination of statistics such as the mean and variance, and of first and second order sensitivity information. Further improvements in computational efficiency are achieved by coupling SRSM with automated source code differentiation tools, which produce code for partial derivatives with respect to model inputs and parameters. One such combination is the SRSM-ADIFOR, a combination of SRSM with the automatic differentiation of FORTRAN (ADIFOR). ADIFOR provides estimates of partial derivatives from a single model run, and this partial derivative information is used in the determination of the coefficients of the polynomial chaos expansions. Furthermore, SRSM can be used in conjunction with Bayesian methods such as Markov chain Monte Carlo (MCMC) methods to reduce uncertainties by incorporating observational information in estimates of model parameters. An overview of the application of SRSM, SRSM-ADIFOR, and the combined SRSM and MCMC methods to complex mechanistic models describing environmental systems is presented here, and the advantages over traditional techniques are discussed.