Polynomial Chaos: Modeling, Estimation, and Approximation

Polynomial chaos decompositions (PCE) have emerged over the past three decades as a standard among the many tools for uncertainty quantification. They provide a rich mathematical structure that is particularly well suited to enabling probabilistic assessments in situations where interdependencies between physical processes or between spatiotemporal scales of observables constitute credible constraints on system-level predictability. Algorithmic developments exploiting their structural simplicity have permitted the adaptation of PCE to many of the challenges currently facing prediction science. These include requirements for large-scale high-resolution computational simulations implicit in modern applications, non-Gaussian probabilistic models, and non-smooth dependencies and for handling general vector-valued stochastic processes. This chapter presents an overview of polynomial chaos that underscores their relevance to problems of constructing and estimating probabilistic models, propagating them through arbitrarily complex computational representations of underlying physical mechanisms, and updating the models and their predictions as additional constraints become known.