Efficient Numerical Algorithms for Solving Nonlinear Filtering Problems
Type of Degreedissertation
Mathematics and Statistics
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Examples of nonlinear filtering problems arise in biology, mathematical finance, signal processing, image processing, target tracking and many engineering applications. Commonly used numerical simulation methods are the Bayesian filter which is derived from the Bayesian formula and the Zakai filter which is related to a system of stochastic partial differential equation known as ``the Zakai equation''. This dissertation mainly focuses on developing and analysing novel, efficient numerical algorithms for solving nonlinear filtering problems. We first introduce a novel numerical algorithm which lies in the general framework on the Bayesian filter. The algorithm is constructed based on samples of the current state obtained by solving the state equation implicitly. We call this algorithm the ``implicit filter method''. Rigorous analysis has been done to prove the convergence of the algorithm. Through numerical experiments we show that our algorithm is more accurate than the Kalman filter and more stable than the particle filter. In the second effort of this work, we propose a hybrid numerical algorithm for the Zakai filter to solve nonlinear filtering problems efficiently. The algorithm combines the splitting-up finite difference scheme and hierarchical sparse grid method to solve moderately high dimensional nonlinear filtering problems. When applying hierarchical sparse grid method to approximate bell-shaped solutions in most applications of nonlinear filtering problem, we introduce a logarithmic approximation to reduce the approximation errors. Some space adaptive methods are also introduced to make the algorithm more efficient. Numerical experiments are carried out to demonstrate the performance and efficiency of our algorithm. In this dissertation, we also develop high order numerical approximation methods for backward doubly stochastic differential equations (BDSDEs). One of the most important properties of BDSDEs is it's equivalence to the Zakai equation. In this connection, our numerical approximation methods for BDSDEs can be considered as efficient numerical approaches to solving nonlinear filtering problems. The convergence order is proved through rigorous error analysis for each algorithm. Numerical experiments are carried out to verify the theoretical results and to demonstrate the efficiency of the proposed numerical scheme.