Advanced Numerical Methods for Computing Statistical Quantities of Interest.

摘要

We have developed novel, effective, robust algorithms that were subjected to mathematically rigorous analyses and extensive computational testing for uncertainty quantification problems of several types that involve high a high number of random variables as inputs to complex systems described by partial differential equations. The approaches developed result in superior means for science-based risk assessment and policy decision making, for simulating fluid flows and structures and other physical systems that are subject to uncertainties and for which the accurate determination of output uncertainties is crucial to the design of devices and strategies in many applications. In our efforts, we have not only availed ourselves of the latest computational and mathematical tools to design our robust, efficient, and accurate methodologies, but we have also invented several new tools for this purpose that are already been used by other researchers. Included in the latter category are multilevel stochastic collocation methods, hyperspherical transformations for discontinuity detection, transformation methods that replace difficult to treat uncorrelated random fields by correlated ones, and hierarchical approaches for improving filtering and Baysean inference problems.