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Wednesday, January 23

New Tools for Uncertainty Quantification and
Data Assimilation in Complex Systems

Tarek Ali El Moselhy, The Massachusetts Institute of Technology, Cambridge
Computer Science and Mathematics Division Seminar
10:00 AM — 11:00 AM, Research Office Building (5700), Room L-204
Contact: Clayton Webster (, 865-574-3649


In this talk, Dr. Tarek Ali El Moselhy will present new tools for forward and inverse uncertainty quantification (UQ) and data assimilation.

In the context of forward UQ, Dr. Moselhy will briefly summarize a new scalable algorithm particularly suited for very high-dimensional stochastic elliptic and parabolic PDEs. The algorithm relies on computing a compact separated representation of the stochastic field of interest. The separated presentation is computed iteratively and adaptively via a greedy optimization algorithm. The algorithm has been successfully applied to problems of flow and transport in stochastic porous media, handling "real world" levels of spatial complexity and providing orders of magnitude reduction in computational time compared to state of the art methods.

In the context of inverse UQ, Dr. Moselhy will present a new algorithm for the Bayesian solution of inverse problems. The algorithm explores the posterior distribution by finding a {\it transport map} from a reference measure to the posterior measure, and therefore does not require any Markov chain Monte Carlo sampling. The map from the reference to the posterior is approximated using polynomial chaos expansion and is computed via stochastic optimization. Existence and uniqueness of the map are guaranteed by results from the optimal transport literature. The map approach is demonstrated on a variety of problems, ranging from inference of permeability fields in elliptic PDEs to benchmark high-dimensional spatial statistics problems such as inference in log-Gaussian cox point processes.

In addition to its computational efficiency and parallelizability, advantages of the map approach include: providing clear convergence criteria and error measures, providing analytical expressions for posterior moments, evaluating at no additional computational cost the marginal likelihood/evidence (thus enabling model selection), the ability to generate independent uniformly-weighted posterior samples without additional model evaluations, and the ability to efficiently propagate posterior information to subsequent computational modules (thus enabling stochastic control).

In the context of data assimilation, Dr. Moselhy will present an optimal map algorithm for filtering of nonlinear chaotic dynamical systems. Such an algorithm is suited for a wide variety of applications including prediction of weather and climate. The main advantage of the algorithm is that it inherently avoids issues of sample impoverishment common to particle filters, since it explicitly represents the posterior as the push forward of a reference measure rather than with a set of samples.

Dr. Tarek Ali El Moselhy will conclude the talk with potential extensions to the proposed algorithms, and a few open questions.