Uncertainty Propagation Using Polynomial Chaos

Sensitivity analysis describes local variation or attributes global variation, but recovering an output distribution by sampling can still require many model evaluations. Polynomial chaos replaces repeated sampling with a spectral representation of the random model response when the uncertain dimension is modest and the response is sufficiently regular.

The treatment assumes probability, linear algebra, and ordinary differential equations. The required Hilbert-space concepts and orthogonal-polynomial constructions are developed locally.

We begin with square-integrable random functions and orthonormal bases, then construct polynomial bases for uniform, Gaussian, and more general input distributions. Tensor products extend the construction to several inputs, and Galerkin projection propagates those inputs through dynamical systems. A final limitations discussion identifies the effects of dimension, irregular responses, and long-time dynamics. Polynomial chaos thereby converts selected uncertainty-propagation problems into deterministic coefficient calculations with directly accessible moments.