  Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty
in models parameterized by independent random variables. The assumption of
independence leads to simple strategies for evaluating PCE coefficients. In
contrast, the application of PCE to models of dependent variables is much more
challenging. Three approaches can be used. The first approach uses mapping
methods where measure transformations, such as the Nataf and Rosenblatt
transformation, can be used to map dependent random variables to independent
ones; however we show that this can significantly degrade performance since the
Jacobian of the map must be approximated. A second strategy is the class of
dominating support methods which build PCE using independent random variables
whose distributional support dominates the support of the true dependent joint
density; we provide evidence that this approach appears to produce
approximations with suboptimal accuracy. A third approach, the novel method
proposed here, uses Gram-Schmidt orthogonalization (GSO) to numerically compute
orthonormal polynomials for the dependent random variables. This approach has
been used successfully when solving differential equations using the intrusive
stochastic Galerkin method, and in this paper we use GSO to build PCE using a
non-intrusive stochastic collocation method. The stochastic collocation method
treats the model as a black box and builds approximations of model output from
a set of samples. Building PCE from samples can introduce ill-conditioning
which does not plague stochastic Galerkin methods. To mitigate this
ill-conditioning we generate weighted Leja sequences, which are nested sample
sets, to build accurate polynomial interpolants. We show that our proposed
approach produces PCE which are orders of magnitude more accurate than PCE
constructed using mapping or dominating support methods.
