Stochastic Partial Differential Equations with Applications to Nonlinear Filtering

小行星9802423 每当一个外野手跟踪并接住一个飞球时，他就直观地解决了一个问题。 目标跟踪问题。这个问题一直困扰着工程师、数学家和 计算机科学家多年来两位伟大的数学家，美国人，诺伯特·维纳和 俄国人安德烈·科尔莫戈罗夫在第二次世界大战期间第一次接触到这个问题。而 而不是接住高飞球，Kolmogorov和Wiener在之前的几天里， 计算机，开发数学算法，这将有助于跟踪敌机， 雷达。Kolmogorov和Wiener发起的研究已发展成为一项蓬勃发展的研究 应用数学领域的过滤理论。 滤波、信号估计 或噪声数据的图像，是目标数据同化的基本组成部分 跟踪.它在导航、图像和信号处理、控制等领域具有重要意义 理论，自动跟踪系统和其他工程和科学领域。这 研究是数学研究所的研究人员之间的联合合作努力 和信息学，维尔纽斯，立陶宛，和研究人员在南方大学 加州。该项目致力于随机偏微分的应用 非线性滤波器研究的重点有两个：（1）非线性滤波中抛物型偏微分方程的Cauchy边值问题 随机过程在某些约束下的演化;（2）非线性滤波 分布式观测及其在图像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 相对简单的非线性滤波算法， 计算和内存要求的在线实施，但几乎 从统计观点来看是最优的，将被开发用于各种各样的应用。 这些应用包括空中交通管制、基于动作捕捉的人机接口以及先进的光学和磁性注册系统。

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.