Collaborative Research: CMG: Uncertainty Quantification in Geophysical State Estimation

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0530858/0530867This collaboration between oceanographers, numerical analysts, and computer scientists is directed at the problems of using extremely large, complex models and data sets. Such problems appear in oceanography, but also in meteorology, economics, engineering, and all fields in which complex simulations are carried out. Automatic differentiation (AD) tools have proved extremely powerful in determining model sensitivities to perturbations in initial and boundary conditions, as well as in internal parameters. AD is critical in recent efforts to bring these models into consistency with modern, massive global data sets (state estimation or "data assimilation" in meteorology). However, sensitivities alone are of limited value for characterizing model behavior. The proposed research on uncertainty quantification represents a qualitative advance in our understanding of the models and will ultimately guide model improvements; this requires new mathematical approaches for eigen-solvers, Hessian computations, and non-smooth optimization to handle the computationally complex models.Intellectual Merit This project seeks to gain deep new insight into geophysical models and the uncertainty inherent in the state estimation of geophysical systems. Without them it is impossible to attribute problems encountered in such models to either non-smooth formulation of the model numerics, or to theoretical limits of the underlying smooth dynamical system. However, significant advances are needed in the algorithms used for uncertainty quantification, which necessitate computing the eigen-solutions of the large, dense Hessians that characterize geophysical models. Substantial advances in automatic differentiation algorithms for computing Hessian-vector products, coupled with novel pre-conditioners based on quasi-Newton updates and scale probing are expected to enable an efficient characterization of the numerical uncertainties. Furthermore, advances in state estimation for highly discontinuous systems, achieved via the use of non-smooth optimization algorithms and corresponding advances in differentiation algorithms, will provide insight into the model uncertainties introduced through the use of non-smooth parameterization schemes. Broader Impacts Numerical models are used in a wide range of scientific and engineering problems, including geophysics; economics; physics; mechanical, nuclear, aeronautical and chemical engineering; and medicine. The complexity of these models, which are typically used in simulation mode, increases over time until no single individual understands how and why the code responds to changes in external or internal parameters. The proposed mathematical algorithms and AD methods will produce sensitivity tests that are computationally feasible even in very large scale problems. Often, the model state simulations are combined with a wide variety of observations so as to produce best estimates of the true state. AD tools have proven extremely useful in enabling the resulting extremely large optimization problem to be solved by using gradient-based optimization algorithms. The proposed work addresses both discontinuities and nonlinearities in large scale models, offering insights that will benefit a wide class of problems where uncertainty quantification has previously been intractable. This project will provide the mathematical and computational tools so that researchers can evaluate model uncertainties in near-automatic fashion. To maximize their availability and impact all algorithms will be implemented as open source software.