Adaptive Polynomial Dimensional Decompositions for Uncertainty Quantification in High Dimensions

The main theme of this paper is intelligently derived truncation strategies for polynomial dimensional decomposition (PDD) of a high-dimensional stochastic response function commonly encountered in engineering and applied sciences. The truncations exploit global sensitivity analysis for defining the relevant pruning criteria, resulting in two new adaptive-sparse versions of PDD: (1) a fully adaptive-sparse approximation, where PDD component functions of an arbitrary number of input variables are retained by truncating the degree of interaction among input variables and the order of orthogonal polynomials according to some specified error tolerance criteria; (2) a partially adaptive-sparse approximation, where PDD component functions with a specified degree of interaction are retained by truncating the order of orthogonal polynomials, fulfilling relaxed error tolerance criteria. The former approximation is rigorous and converges to the exact second-moment statistics of a stochastic response when the error tolerances vanish. The latter approximation, although obtained less rigorously than the former approximation, is expected to solve practical engineering problems with numerous variables. Compared with past developments, the adaptive PDD methods do not require its truncation parameter(s) to be assigned a priori or arbitrarily. The computational complexity of the adaptive PDD remains polynomial, as opposed to exponential, mitigating the curse of the dimensionality to some extent. Numerical examples are presented to demonstrate the usefulness of the proposed methods for uncertainty quantification in complex systems.