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Projection onto polynomial chaos (PC) basis functions is often used to reformulate a system of ordinary differential equations (ODEs) with uncertain parameters and initial conditions as a deterministic ODE system that describes tide evolution of the PC modes. The deterministic Jacobian of this projected system is different and typically much larger than the random Jacobian of tide original ODE system. This paper shows that the location of the eigenvalues of the projected Jacobian is largely determined by the eigenvalues of the original Jacobian, regardless of PC order or choice of orthogonal polynomials. Specifically, the eigenvalues of the projected Jacobian always lie in tide convex hull of the numerical range of the Jacobian of the original system. [ABSTRACT FROM AUTHOR]

Abstract: A critical problem in accurately estimating uncertainty in model predictions is the lack of details in the literature on the correlation (or full joint distribution) of uncertain model parameters. In this paper we describe a framework and a class of algorithms for analyzing such “missing data” problems in the setting of Bayesian statistics. The analysis focuses on the family of posterior distributions consistent with given statistics (e.g. nominal values, confidence intervals). The combining of consistent distributions is addressed via techniques from the opinion pooling literature. The developed approach allows subsequent propagation of uncertainty in model inputs consistent with reported statistics, in the absence of data. [Copyright &y& Elsevier]