RUI: Efficient Adaptive Backward Stochastic Differential Equation Methods for Nonlinear Filtering Problems

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Nonlinear filtering problem is a mathematical model for system estimation in signal processing problems arising from various scientific and engineering fields. Examples of the nonlinear filter's applications include tracking an aircraft using radar measurements, estimating a digital communications signal using noisy measurements, and estimating the volatility of financial instruments using stock market data. The key mission of the nonlinear filtering problem is to establish a "best estimate" for the true value of a dynamic system from an incomplete, potentially noisy set of observations on that system. The goal of this project is to develop novel numerical algorithms, which are accurate and efficient for the nonlinear filtering problem, by solving a backward stochastic differential equation (SDE) system. The proposed project will engage undergraduate students at an RUI institution in computational and applied mathematics research.The cornerstone of this proposed approach, named the backward SDE filter, is the fact that the solution of the backward SDE system is the probability density function of the signal state as required in the nonlinear filtering problem. This project will start with the construction of backward SDE filter algorithms that are high order in time and adaptive in space, which blends the strengths of well known methods from this area of research. Then, the applicability of the backward SDE filter will be enlarged to tackle the grand challenge problems. Specifically, massively parallel algorithms will be designed for the backward SDE filter so that it could be implemented to solve large scale scientific computing problems on high performance computing facilities. The backward SDE filter is a new approach to solve the nonlinear filtering problem, and it addresses the main issues in the numerical solutions for nonlinear filtering problems, such like the low regularity problem and the high dimensionality problem. As a result, the backward SDE filter will provide scientists and engineers in various disciplines an accurate, efficient, and easy to use algorithm for data assimilation.