Stochastic Partial Differential Equations with Applications to Nonlinear Filtering

每次外野手追踪并接住一个飞球时，他都直观地解决了一个目标跟踪问题。这个问题多年来一直困扰着工程师、数学家和计算机科学家。两位伟大的数学家，美国的诺伯特·维纳和俄国的安德烈·柯尔莫哥罗夫，在第二次世界大战期间首次研究了这个问题。在计算机出现之前的日子里，柯尔莫哥洛夫和维纳试图开发数学算法，以帮助通过雷达跟踪敌机，而不是捕捉飞球。由Kolmogorov和Wiener发起的这项研究已经发展成为应用数学的一个蓬勃发展的领域，即过滤理论。滤波，即从噪声数据中估计信号或图像，是目标跟踪中数据同化的基本组成部分。它在导航、图像和信号处理、控制理论、自动跟踪系统以及其他工程和科学领域具有核心重要性。这项研究是立陶宛维尔纽斯数学与信息学研究所的研究人员和南加州大学的研究人员共同合作的成果。本课题主要研究随机偏微分方程在非线性滤波中的应用。研究的重点有两个方面：(1)在一定约束下演化的随机过程的非线性滤波中出现的抛物型偏微分方程的Cauchy边界问题；(2)分布式观测非线性滤波及其在图像中低可观测目标跟踪中的应用。研究还将考虑非线性滤波的数值方面。预计将为各种各样的应用开发相对简单的非线性滤波算法，这些算法对在线实现的计算和内存要求不太高，但从统计的角度来看几乎是最佳的。其应用包括空中交通管制、基于动作捕捉的人机界面以及先进的光学和磁登记系统。

9802423 Rozovskii Each time an outfielder tracks and catches a fly ball, he intuitively solves a problem of target tracking. That problem has stymied engineers, mathematicians, and computer scientists for years. Two great mathematicians, American, Norbert Wiener and Russian, Andrey Kolmogorov, first approached the problem during World War II. Rather than catching fly balls, Kolmogorov and Wiener were trying, in the days before computers, to develop mathematical algorithms that would help to track enemy aircraft by radar. The research started by Kolmogorov and Wiener has developed into a thriving area of applied mathematics known as Filtering Theory. Filtering, estimation of a signal or an image from noisy data, is the basic component of the data assimilation in target tracking. It is of central importance in navigation, image and signal processing, control theory, automatic tracking systems and other areas of engineering and science. This research is a joint collaborative effort between researchers at the Institute of Mathematics and Informatics, Vilnius, Lithuania, and researchers at the University of Southern California. The project is devoted to applications of stochastic partial differential equations to nonlinear filtering. The focus of the research is twofold: (1) Cauchy- boundary problems for parabolic partial differential equations arising in nonlinear filtering of stochastic processes evolving under some constraints; and (2) nonlinear filtering with distributed observation and applications to tracking of low-observable targets in images. The research will also consider the numerical aspects of nonlinear filtering. It is expected that relatively simple nonlinear filtering algorithms which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal from a statistical viewpoint will be developed for a wide variety of applications. The applications include air traffic control, human-computer interfaces based on motion- capture, and advanced optical and magnetic registration systems.