Adaptive and high order PDF methods for nonlinear filtering problems

本研究项目研究非线性滤波问题的高效快速数值算法。非线性滤波的目的是基于带噪声的部分状态观测，获得随机动力系统状态的最优统计估计，该状态可以是无人机的轨迹。作为数据同化的重要工具，非线性滤波在生物学、数学金融、信号处理和目标跟踪等领域有着广泛的应用。目标跟踪的一个具体例子是无人机的制导和监视，无人机在国家安全和国防中发挥着至关重要的作用。通过基于多个传感器的观测数据对目标位置进行非线性估计，非线性滤波成为无人机目标定位和导航系统中的核心部件。由于海量数据的可用性不断提高，传统的非线性滤波方法如卡尔曼滤波和粒子滤波往往不能很好地处理高维、高度非线性的问题。本研究项目旨在开发快速、自适应的数值算法来解决此类非线性滤波问题。几个研究生将参与这个项目。本项目是研究开发高阶和自适应的数值算法，以及相应的非线性滤波问题的数值分析。重点研究了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：第一类算法在自适应构造的计算区域和稀疏网格上求解Zakai方程；第二类算法求解一类倒向随机微分方程。这两种算法都克服了PDF滤波的三个困难：i)高维；ii)低正则性；iii)无界域。这项建议的主要目标是使PDF过滤器成为一种具有很强竞争力的非线性过滤问题的数值方法。特别是，本文研究的基于BSDE的新型PDF滤波器达到了高阶收敛，比现有的所有非线性滤波方法都要快得多。

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.