Adaptive and high order PDF methods for nonlinear filtering problems

本研究计画探讨非线性滤波问题之快速有效数值演算法。 非线性滤波的目标是基于状态的噪声部分观测值获得随机动力系统的状态的最佳统计估计，该状态可以是无人机的轨迹。 作为数据同化的一个重要工具，非线性滤波在生物学、数学金融、信号处理和目标跟踪等广泛的研究领域有着广泛的应用。目标跟踪的一个具体例子是无人机（UAV）的引导和监视，无人机在国家安全和国防中发挥着重要作用。 非线性滤波是根据多个传感器的观测值对目标位置进行非线性估计的一种方法，是无人机目标定位和导航系统的核心组成部分。 由于数据量巨大且可用率不断提高，传统的非线性滤波方法如卡尔曼滤波、粒子滤波等往往不足以处理高维、高度非线性的问题。 本研究计画旨在发展快速且自适应的数值演算法，以解决此类非线性滤波问题。几个研究生将参加这个项目。本计画主要针对非线性滤波问题，进行高阶与自适应数值演算法之发展与数值分析之研究。 重点是PDF滤波器，它通过随机偏微分方程或倒向随机微分方程求解最优滤波器的条件概率密度函数（PDF）来解决非线性滤波问题。 我们将开发两类算法：第一个解决自适应构造的计算域和稀疏网格上的Zakai方程;第二个解决一类向后随机微分方程。这两种算法都克服了PDF滤波器的三个困难：i）高维性; ii）低正则性; iii）无界域。这个建议的首要目标是使PDF滤波器的非线性滤波问题的一个非常有竞争力的数值方法。特别是，新的基于Bundle的PDF滤波器研究在这个建议中实现了高阶收敛，这是远远快于所有其他现有的非线性滤波方法。

This research project investigates efficient and fast numerical algorithms of nonlinear filtering problems. The goal of nonlinear filtering is to obtain best statistical estimate of the state, which can be the trajectory of an unmanned aerial vehicle, of a stochastic dynamical system based on noisy partial observations of the state. As a key tool for data assimilation, nonlinear filtering has applications in vastly diverse research areas including biology, mathematical finance, signal processing and target tracking. A particular example of target tracking is the guidance and surveillance of unmanned aerial vehicles (UAV) which have been playing an essential role in national security and defense. Through nonlinear estimates of target location based on the observations from multiple sensors, nonlinear filtering forms a core component in unmanned aerial vehicle targeting and navigation systems. Because of vast amount data at an increasing rate of availability, traditional nonlinear filtering methods such as Kalman filter and particle filter are often inadequate to handle high dimensional and highly nonlinear problems. This research project aims to develop fast and adaptive numerical algorithms to attack such nonlinear filtering problems. Several graduate students will participate in this project. This project is to conduct research in developing high order and adaptive numerical algorithms and the corresponding numerical analysis on nonlinear filtering problems. The focus is on the PDF filter, which solves the nonlinear filtering problem by solving the conditional probability density function (PDF) for the optimal filter through a stochastic partial differential equation or a backward stochastic differential equation. We will develop two classes of algorithms: the first solves the Zakai equation on adaptively constructed computational domains and on sparse grids; and the second solves a class of backward stochastic differential equations. Both algorithms overcome three difficulties for the PDF filter: i) high dimensionality; ii) low regularity; iii) unbounded domains. The overarching objective of this proposal is to make the PDF filter a highly competitive numerical method for nonlinear filtering problems. In particular, the novel BSDE based PDF filter studied in this proposal achieves a high order of convergence, which is much faster than all other existing nonlinear filtering methods.