Mathematical and Computational Tools for Predictive Simulation of Complex Coupled Systems under Uncertainty

Methods and algorithms are developed to enable the accurate analysis of problems that exhibit interacting physical processes with uncertainties. These uncertainties can pertain either to each of the physical processes or to the manner in which they depend on each others. These problems are cast within a polynomial chaos framework and their solution then involves either solving a large system of algebraic equations or a high dimensional numerical quadrature. In both cases, the curse of dimensionality is manifested. Procedures are developed for the efficient evaluation of the resulting linear equations that advantage of the block sparse structure of these equations, resulting in a block recursive Schur complement construction. In addition, embedded quadratures are constructed that permit the evaluation of very high-dimensional integrals using low-dimensional quadratures adapted to particular quantities of interest. The low-dimensional integration is carried out in a transformed measure space in which the quantity of interest is low-dimensional. Finally, a procedure is also developed to discover a low-dimensional manifold, embedded in the initial high-dimensional one, in which scalar quantities of interest exist. This approach permits the functional expression of the reduced space in terms of the original space, thus permitting cross-scale sensitivity analysis