Paul Constantine

Paul Diaz, Paul Constantine, Kelsey Kalmbach et al.

A modified, deterministic SEIR model is developed for the 2014 Ebola epidemic occurring in the West African nations of Guinea, Liberia, and Sierra Leone. The model describes the dynamical interaction of susceptible and infected populations, while accounting for the effects of hospitalization and the spread of disease through interactions with deceased, but infectious, individuals. Using data from the World Health Organization (WHO), parameters within the model are fit to recent estimates of infected and deceased cases from each nation. The model is then analyzed using these parameter values. Finally, several metrics are proposed to determine which of these nations is in greatest need of additional resources to combat the spread of infection. These include local and global sensitivity metrics of both the infected population and the basic reproduction number with respect to rates of hospitalization and proper burial.

Paul Constantine, Michael Emory, Johan Larsson et al.

We present a computational analysis of the reactive flow in a hypersonic scramjet engine with focus on effects of uncertainties in the operating conditions. We employ a novel methodology based on active subspaces to characterize the effects of the input uncertainty on the scramjet performance. The active subspace identifies one-dimensional structure in the map from simulation inputs to quantity of interest that allows us to reparameterize the operating conditions; instead of seven physical parameters, we can use a single derived active variable. This dimension reduction enables otherwise infeasible uncertainty quantification, considering the simulation cost of roughly 9500 CPU-hours per run. For two values of the fuel injection rate, we use a total of 68 simulations to (i) identify the parameters that contribute the most to the variation in the output quantity of interest, (ii) estimate upper and lower bounds on the quantity of interest, (iii) classify sets of operating conditions as safe or unsafe corresponding to a threshold on the output quantity of interest, and (iv) estimate a cumulative distribution function for the quantity of interest.

Paul Constantine, David Gleich

Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace methods are the eigenvectors of a symmetric, positive semidefinite matrix whose elements are the average products of partial derivatives of the simulation's input/output map. We study a Monte Carlo method for approximating the eigenpairs of this matrix. We offer both theoretical results based on recent non-asymptotic random matrix theory and a practical approach based on the bootstrap. We extend the analysis to the case when the gradients are approximated, for example, with finite differences. Our goal is to provide guidance for two questions that arise in active subspaces: (i) How many gradient samples does one need to accurately approximate the eigenvalues and subspaces? (ii) What can be said about the accuracy of the estimated subspace, both theoretically and practically? We test the approach on both simple quadratic functions where the active subspace is known and a parameterized PDE with 100 variables characterizing the coefficients of the differential operator.