In this project, the investigators and their colleagues deepen and extend their asymptotic analysis of data assimilation algorithms for systems of nonlinear partial differential equations in the presence of measurement error and stochastic perturbation. This research includes the development of a stochastic convergence theory for ensemble and particle filters in infinite-dimensional Banach spaces, a corresponding statistical theory for finite-dimensional asymptotics, and a study of the behavior of spectral approximations for the spatio-temporal covariance structures of those filters. Of special importance is the computational efficiency and statistical convergence of wavelet and FFT spectral approximations in very-high-dimensional non-Gaussian cases. Connections between probability measures on Sobolev spaces, random fields in spatial statistics, and stochastic spectral expansions are exploited to effectively reduce the dimensionality of the system. The goal is to develop and demonstrate very fast and memory-efficient algorithms that provably converge in a suitable stochastic sense to the correct answer. Using high-performance computing facilities, the investigators' model of the spread of wildland fires, expressed as a coupled weather-fire system driven by real-world or Monte Carlo data, serves as the primary testbed for assessing the computational efficiency and statistical convergence properties of a wide variety of data assimilation algorithms and approximation methods.
Data assimilation is the art and science of incorporating real-time information, as it arrives, into a running complex simulation, in such a way that the simulation adjusts and adapts in a robust and reasonable way to the new data. The subject area of greatest interest to this project is the tracking of wildland fires as they spread across extended terrains that can include forests, grasslands, and human communities. For this purpose the investigators maintain a wildland fire-and-weather simulator that runs in a high-performance computing facility, embedded in a data assimilation framework so that the model can correct itself in response to incoming data (overhead photographs, weather station data, etc) as it arrives. The general methodology of data assimilation is of interest to many areas of science, and so it is critical that the algorithms employed be computationally very efficient and statistically robust, and that they can be proven to converge in the appropriate limit to the right answer. This project is dedicated to the development and study of such algorithms, and to rigorous analytical proofs of convergence and efficiency. Doctoral students and post-doctoral associates involved in the project receive a highly interdisciplinary exposure to computational and applied mathematics, statistics, meteorology, atmospheric physics, and fire science.
