Quantification of Uncertainty: Previous, Current, and Future Directions

My general research areas are sensitivity analysis and computational error estimation for numerical solutions of differential equations. Most of my research focuses on inverse problems with the specific goal of quantifying the uncertainty of input parameters given uncertain output data. I obtained my thesis at Colorado State University under the supervision of Dr. Don Estep where we developed a computational measure-theoretic method to propagate probability distributions on output data through response surface approximations to obtain a probability measure on input parameter space [4, 3, 10, 9, 11, 12]. This research is currently being applied to real-world problems involving storm surge (supported by a grant from the NSF) and subsurface contaminant transport (supported by a grant from the DOE). Since obtaining my Ph.D., I have engaged in several multidisciplinary collaborations involving research in uncertainty quantification using methods including, but not limited to, surrogate response surfaces based on stochastic finite elements/polynomial chaos and data assimilation techniques to improve the forecasting of storm surge resulting from hurricanes and tropical cyclones. I am currently a co-principal investigator for the project Data-Driven Inverse Sensitivity Analysis for Predictive Coastal Ocean Modeling funded by the National Science Foundation’s Division of Mathematical Sciences. As an ICES (Institute for Computational Engineering and Sciences) Postdoctoral Fellow and subsequently as a Research Associate at The University of Texas at Austin, I collaborated with Dr. Tim Wildey and Dr. Clint Dawson to study various aspects of uncertainty quantification using polynomial chaos expansions. In [7], we develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. The work of [7] is currently being extended by two groups of researchers that I collaborate with at both The University of Texas at Austin and at the Sandia National Laboratory in Albuquerque. In [6], we develop computable a posteriori error estimates for the point-wise evaluation of linear functionals of a solution to a parameterized linear system of equations. These error estimates are based on a variational analysis applied to polynomial spectral methods for forward and adjoint problems. We also use this error estimate to define an improved linear functional and we prove that this improved functional converges at a much faster rate than the original linear functional given a point-wise convergence assumption on the forward and adjoint solutions. The advantage of this method is that we are able to use low order spectral representations for the forward and adjoint systems to cheaply produce linear functionals with the accuracy of a higher order spectral representation. In [8], we explore the effect of numerical error on the propagation of distributions for both the forward problem where the distribution of input parameters is known, and also the inverse problem solved using a Bayesian framework. Specifically, we develop computable bounds on the error in computed distribution functions for both the forward and inverse problem. Dr. Wildey and I co-organized the minisymposium “A Posteriori Error Estimation for Reliable Uncertainty Quantification” for the 2012 SIAM Conference on Uncertainty Quantification to bring together research scientists studying topics related to these works. We are currently exploring the use of the adjoint based a posteriori error estimates to improve