Nonlinear filtering of high dimensional, chaotic, multiple timescale correlated systems

This dissertation addresses theoretical and numerical questions in nonlinear filtering theory for high dimensional, chaotic, multiple timescale correlated systems. The research is motivated by problems in the geosciences, in particular oceanic or atmospheric estimation and climate prediction. As the capability and need to further resolve the physics models on finer scales continues, greater spatial and temporal scales become present and the dimension of the models becomes increasingly large. In the atmospheric sciences, these models can be of the order $\mathcal{O}(10^9)$ degrees of freedom and require assimilation of the order $\mathcal{O}(10^7)$ observations during a single day. The models are chaotic and the observing sensors may be correlated with the physical processes themselves. The goal of the dissertation is to develop theoretical results that can provide the mathematical justification for new filtering algorithms on a lower dimensional problem, and to develop novel methods for dealing with issues that plague particle filtering when applied to high dimensional, chaotic, multiple timescale correlated systems. The first half of the dissertation is theoretical and addresses the question of approximating the continuous time nonlinear filtering equation for a multiple timescale correlated system by an averaged filtering equation in the limit of large timescale separation. The first result in this direction is within the context of a slow-fast system with correlation between the slow process and the observation process, and when we are only interested in estimating functions of the slow process. The main result is that we can retrieve a rate of convergence and that there is a metric generating the topology of weak convergence, such that the marginal filter converges to the averaged filter at the given rate in the limit of large timescale separation. The proof uses a probabilistic representation (backward doubly stochastic differential equation) of the dual process to the unnormalized filter, and sharp estimates on the transition density and semigroup of the fast process. The second theoretical result of the dissertation addresses the same question for a broader problem, where the slow signal dynamics include an intermediate timescale forcing. We prove that the marginal filter converges in probability to the average filter for a metric that generates the topology of weak convergence. The method of proof is by showing tightness of the measure-valued process, characterizing the weak limits, and proving the limit is unique. The perturbation test function (also known as method of corrector) is used to deal with the intermediate timescale forcing term, where the corrector is the solution of a Poisson equation. The second half of the dissertation develops filtering algorithms that leverage the theoretical results from the first half of the thesis to produce particle filtering methods for the averaged filtering equation. We also develop particle methods that address the issue of particle collapse for filtering on general high dimensional chaotic systems. Using the two timescale Lorenz 1996 atmospheric model, we show that the reduced order particle filtering methods are shown to be at least an order of magnitude faster than standard particle methods. We develop a method for particle filtering when the signal and observation processes are correlated. We also develop extensions to controlled optimal proposal particle filters that improve the diversity of the particle ensemble when tested on the Lorenz 1963 model. In the last chapter of the dissertation, we adopt a dynamical systems viewpoint to address the issue of particle collapse. This time the goal is to exploit the chaotic properties of the system being filtered to perform assimilation in a lower dimensional subspace. A new approach is developed which enables data assimilation in the unstable subspace for particle filtering. We introduce the idea of future right-singular vectors to produce projection operators, enabling assimilation in a lower dimensional subspace. We show that particle filtering algorithms using dynamically generator projection operators, in particular the future right-singular vectors, outperforms standard particle methods in terms of root-mean-square-error, diversity of the particle ensemble, and robustness when applied to the single timescale Lorenz 1996 model

An Automatic Medium to High Fidelity Low-Thrust Global Trajectory Toolchain; EMTG-GMAT

Solving the global optimization, low-thrust, multiple-flyby interplanetary trajectory problem with high-fidelity dynamical models requires an unreasonable amount of computational resources. A better approach, and one that is demonstrated in this paper, is a multi-step process whereby the solution of the aforementioned problem is solved at a lower-fidelity and this solution is used as an initial guess for a higher-fidelity solver. The framework presented in this work uses two tools developed by NASA Goddard Space Flight Center: the Evolutionary Mission Trajectory Generator (EMTG) and the General Mission Analysis Tool (GMAT). EMTG is a medium to medium-high fidelity low-thrust interplanetary global optimization solver, which now has the capability to automatically generate GMAT script files for seeding a high-fidelity solution using GMAT's local optimization capabilities. A discussion of the dynamical models as well as thruster and power modeling for both EMTG and GMAT are given in this paper. Current capabilities are demonstrated with examples that highlight the toolchains ability to efficiently solve the difficult low-thrust global optimization problem with little human intervention