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

非线性滤波问题是信号处理中系统估计的一个数学模型，它产生于各种科学和工程领域。非线性滤波器的应用实例包括使用雷达测量来跟踪飞机、使用噪声测量来估计数字通信信号以及使用股票市场数据来估计金融工具的波动性。非线性滤波问题的关键使命是从一个不完整的、潜在噪声的动态系统的观测值集合中建立一个对该系统的真值的“最佳估计”。本计画的目标是发展新的数值演算法，借由求解倒向随机微分方程系统，精确且有效地解决非线性滤波问题。该项目将邀请RUI研究所的本科生进行计算和应用数学研究。该方法的基础称为后向滤波器，其基础是后向滤波系统的解是非线性滤波问题中所需的信号状态的概率密度函数。该项目将从构建时间上高阶和空间上自适应的后向滤波器算法开始，该算法融合了该研究领域中众所周知的方法的优点。然后，后向滤波器的适用性将被扩大，以解决大的挑战性问题。具体而言，大规模并行算法将被设计为向后的递归滤波器，以便它可以实现解决大规模的科学计算问题的高性能计算设施。后向滤波是一种新的非线性滤波方法，它解决了非线性滤波问题数值解中的低正则性和高维性问题。因此，后向滤波器将为各个学科的科学家和工程师提供一种准确，高效和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.