Developments in Computational Methods for Efficient Uncertainty Analysis and Reduction
S. Balakrishnan, S. Isukapalli, and P.G. Georgopoulos (EOHSI, UMDNJ - R.W. Johnson Medical School and Rutgers University)
Motivation: Comprehensive uncertainty analyses of complex mechanistic models of environmental and biological systems, and uncertainty reduction via incorporation of observational information are essential, but often not feasible. Traditional Monte Carlo methods, e.g. involving standard or Latin Hypercube sampling, for propagating uncertainty and developing probability densities of model outputs, may require performing a large number of model simulations. Similarly, application of Bayesian techniques for incorporating observational information in estimates of model parameters require a large number of model simulations. These approaches, however, can be prohibitive in the case of 3D dynamic finite element or finite difference models, which typically reduce to the numerical solution of millions of simultaneous ordinary differential equations.
Novel Approaches: An alternative approach is provided by the Stochastic Response Surface Method (SRSM), which is a computationally efficient technique that facilitates uncertainty analysis through the determination of statistically equivalent reduced models. Furthermore, SRSM can be combined with Markov Chain Monte Carlo (MCMC) methods to estimate the uncertainties.
Method Details: The 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 like the mean and variance, and of first and second order sensitivity information. Additionally, this polynomial form can be used as a reduced model for performing Bayesian analysis.
Results and Discussion: