In sequential data assimilation problems, the Kalman filter (KF) is optimal for linear Gaussian models. The ensemble Kalman filter (EnKF) has been widely used as an numerical approximation of the KF primarily due to its ease of implementation by Monte Carlo methods. Similarly, the optimal filter/Bayesian filter plays an important role in nonlinear non-Gaussian models. And the particle filter (PF) is widely used as a Monte Carlo version of the Bayesian filter for practical reasons. This thesis consists of two main components: (1) conducting error analysis on EnKF and PF, and (2) proposing new data assimilation methods to improve efficiency and accuracy.

A rigorous analysis on the numerical errors of the EnKF is conducted in a general setting. Error bounds are provided and convergence of the EnKF to the exact Kalman Filter is established. The analysis reveals that the ensemble errors induced by the Monte Carlo sampling can be dominant, compared to other errors such as the numerical integration error of the underlying model equations.

A new error analysis is conducted for PF from the numerical analysis perspective, which is different from the existing convergence results for PF in the probability literature. In this thesis, we demonstrate that the difference between the optimal filter and PF, in general, will increase exponentially. This error mainly consists of two parts: (1) the numerical errors from solving the dynamic equations, and (2) the sampling errors introduced by generating particles. The convergence of the PF to the optimal filter in the weak sense with the numerical error incorporated is proved. And the bounds for one-step local error and cumulative global error are provided.

Both the analysis on EnKF and PF suggest a less obvious fact—more frequent data assimilation may lead to larger numerical errors of the EnKF and PF.

Based on the analysis, two sets of methods to reduce sampling errors for EnKF are developed. First, we present a deterministic sampling strategy(qEnKF) based on cubature rules with much improved accuracy. Second, we propose an efficient EnKF implementation via generalized polynomial chaos (gPC) expansion. The key ingredients of the gPC-based approach involve (1) solving the system of stochastic state equations via the gPC methodology to gain efficiency; (2) sampling the gPC approximation of the stochastic solution with an arbitrarily large number of samples, at virtually no additional computational cost, to drastically reduce sampling errors. The resulting algorithms thus achieve high accuracy at reduced computational cost, compared to the classical implementations of EnKF.

Numerical examples are provided to verify the theoretical findings and to demonstrate the improved performance of the qEnKF and gPC-based EnKF.