| In this thesis, we study how to efficiently solve nonlinear filtering and smoothing problems and joint state and parameters estimation problems. Data assimilation is one of the type of processes that we will be addressing as well since our approach encompasses a method for combining observations of variables into models. Also, our proposed methods can be applied for uncertainty quantification [20, 31]. Our objective is to develop smoothing and filtering algorithm for a large scale model dynamics, especially system governed by large scale partial differential equations dynamics. These large scale dynamics are commonly used in mathematical modeling, data assimilation and uncertainty quantification and have a lot of real life applications. Developing effective and efficient algorithm is essential due to the large scale and complex dynamics. This is why we must develop and analyze efficient but effective filtering algorithm to perform the task at hand.;Our approach is based on the optimal filtering theory; i.e., the optimal filter based on the Bayes' formula for discrete time dynamics and the Zakai equation for continuous time. After understanding the relationship between the discrete time and continuous time filter, we conclude that the discrete time filter with properly determined one step solution map can be applied directly to the continuous time filtering.;Well known result of Bayes' optimal filter is the Kalman filter for linear and Gaussian system. And our objective is to use Gaussian filter for nonlinear (significantly) system to improve the Kalman filter (extended). That is, we develop the filtering update via the assumed Gaussian density filter. A key step is that we update the covariance in the square root factors form and thus we update the square root factors of the Gaussian covariance. This evolves into the reduced Gaussian filter based on the reduced factor updates.;For dissipative system, we also develop an alternative to the reduced Gaussian filter, by the assumed covariance filter. For systems that are time reversible, we use the time reversal filter. As a result we obtain the forward and backward filter for time reversible systems. We also focus on the joint state-parameters estimate for parameters dependent problems such as media identification. |
