Nonlinear filtering and evolution equations: Fast algorithms with applications to target tracking.

摘要

The nonlinear stochastic filtering problem and related parabolic equations with variable coefficients are studied in this dissertation. The main focus is numerical approximation, algorithm development, and applications in target tracking.; After a general introduction of nonlinear filtering and its development, the theory of optimal nonlinear filtering is summarized for all the discrete time, continuous-discrete time, and continuous time models, and a general error estimate for the approximation of the optimal filter in continuous-discrete time is given. Then the approximation of general linear parabolic equations is reviewed in the framework of the method of lines (spatial-temporal discretization). A new “implicit-upwind and explicit-downwind” scheme, a related ADI scheme, and a parallel algorithm are proposed and they are shown to be absolutely stable and second-order accurate in both space and time. Two kinds of operator-splitting methods for the matrix exponential, totaling 16 schemes (including ADI or fractional steps schemes), are generalized to the case of semigroups, generated by unbounded operators in function spaces. Their respective first-order and second-order convergence rates are proved in the general setting. Results for semigroup decompositions of the first kind are then applied to the Fokker-Planck, Kushner, and Zakai equations which need to be solved in the optimal nonlinear filter. These convection-diffusion splitting procedures serve as the basis of the real-time nonlinear filters to be developed in this dissertation.; Two fast algorithms are presented for the continuous-discrete filtering model by using a domain pursuit (or windowing) technique: an explicit scheme with forward characteristics and fast Gauss transform, and an implicit scheme with backward characteristics and ADI method. In both cases, the domain of interest is determined adaptively. A preliminary investigation of multi-resolution wavelet expansion of the unnormalized filtering densities is also conducted in the spirit of local grid refinement.; The last two chapters of the dissertation are devoted to the application of nonlinear filtering to target tracking. Basic mathematical models for maneuvering targets are developed, and practical examples in three and six dimensions with rather satifactory numerical results are presented.