Supervisor: Alan S. Willsky, Professor of Electrical Engineering
Statistical modeling and estimation of large-scale dynamic systems is important in a wide range of scientific applications. Conventional optimal estimation methods, however, are impractical due to their computational complexity. In this thesis, we consider an alternative multiscale modeling framework first developed by Basseville, Willsky, et al.
This multiscale estimation methodology has been successfully applied to a number of large-scale static estimation problems, one of which is the application of the so-called 1/f multiscale models to the mapping of ocean surface height from satellite altimetric measurements. A modified 1/f model is used in this thesis to jointly estimate the surface height of the Mediterranean Sea and the correlated component of the measurement noise in order to remove the artifacts apparent in maps generated with the more simplistic assumption that the measurement noise is white.
The main contribution of this thesis is the extension of the multiscale framework to dynamic estimation. We introduce a recursive procedure that propagates a multiscale model for the estimation errors in a manner analogous to, but more efficient than, the Kalman filter's propagation of the error covariances. With appropriately chosen multiscale models, such as the new class of non-redundant models that we introduce, the computational gain can be substantial. We use 1-D and 2-D diffusion processes to illustrate the development of our algorithm. The resulting multiscale estimators achieve O(N) computational complexity with near-optimal performance in 1-D and O(N^(3/2)) in 2-D, as compared to the O(N^3) complexity of the standard Kalman filter.