Adaptive and high order PDF methods for nonlinear filtering problems

用于非线性滤波问题的自适应和高阶 PDF 方法

• 批准号: 1620150
• 负责人: Yanzhao Cao
• 金额: $ 16万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620150&HistoricalAwards=false
• 关键词: Adaptive; order; PDF; methods; nonlinear

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.

该研究项目研究非线性滤波问题的高效快速数值算法。 非线性滤波的目标是基于状态的噪声部分观测来获得随机动力系统的状态的最佳统计估计，该状态可以是无人驾驶飞行器的轨迹。 作为数据同化的关键工具，非线性滤波在生物学、数学金融、信号处理和目标跟踪等众多研究领域都有应用。目标跟踪的一个具体例子是无人机（UAV）的制导和监视，无人机在国家安全和国防中发挥着重要作用。 通过基于多个传感器的观测对目标位置进行非线性估计，非线性滤波形成了无人机瞄准和导航系统的核心组件。 由于数据量巨大且可用性不断提高，传统的非线性滤波方法（例如卡尔曼滤波器和粒子滤波器）通常不足以处理高维和高度非线性问题。 该研究项目旨在开发快速自适应数值算法来解决此类非线性滤波问题。几名研究生将参与该项目。该项目旨在研究开发高阶自适应数值算法以及非线性滤波问题的相应数值分析。 重点是PDF滤波器，它通过随机偏微分方程或向后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：第一类在自适应构造的计算域和稀疏网格上求解 Zakai 方程；第二类在自适应构造的计算域和稀疏网格上求解 Zakai 方程。第二个求解一类向后随机微分方程。两种算法都克服了PDF过滤器的三个困难：i）高维； ii) 低规律性； iii) 无界域。该提案的总体目标是使 PDF 滤波器成为解决非线性滤波问题的极具竞争力的数值方法。特别是，本提案中研究的基于 BSDE 的 PDF 滤波器实现了高阶收敛，比所有其他现有的非线性滤波方法快得多。