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

非线性滤波问题是在各种科学和工程领域中出现的信号处理问题中系统估计的数学模型。非线性滤波器的应用实例包括使用雷达测量跟踪飞机，使用噪声测量估计数字通信信号，以及使用股票市场数据估计金融工具的波动性。非线性滤波问题的关键任务是从对该系统的不完整的、可能有噪声的一组观测值中建立对该动态系统真值的“最佳估计”。本课题的目标是通过求解一个倒向随机微分方程（SDE）系统，开发一种新的精确、高效的非线性滤波数值算法。该项目将招收芮学院计算与应用数学研究专业的本科生。该方法被称为后向SDE滤波器，其基础是后向SDE系统的解是非线性滤波问题中所要求的信号状态的概率密度函数。本项目将从构建反向SDE滤波算法开始，该算法在时间上是高阶的，在空间上是自适应的，它融合了该研究领域中已知方法的优势。然后，扩大后向SDE滤波器的适用性，以解决大挑战问题。具体而言，将为后向SDE滤波器设计大规模并行算法，使其能够在高性能计算设备上实现解决大规模科学计算问题。后向SDE滤波器是解决非线性滤波问题的一种新方法，它解决了非线性滤波问题数值解中的主要问题，如低正则性问题和高维数问题。因此，后向SDE滤波器将为各学科的科学家和工程师提供一种准确、高效、易于使用的数据同化算法。

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.