Diffusion Processes and Partial Differential Equations

扩散过程和偏微分方程

• 批准号: 0653121
• 负责人: Nicolai Krylov
• 金额: $ 48.58万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653121&HistoricalAwards=false
• 关键词: Diffusion; Processes; Partial; Differential; Equations

Diffusion Processes and Partial Differential equations Abstract of Proposed Research Nicolai V KrylovThe project relates to some important modern topics in the theories of Diffusion Processes and Partial Differential Equations (PDEs). They arise from practical applications and include optimal control of random processes, optimal filtering of diffusion processes, white noise driven stochastic PDEs (SPDEs) arising in population genetics, the theory of fully nonlinear PDEs. The project includes an investigation of the numerical methods of finding solutions.Fully nonlinear PDEs arise in the optimal mass transportation problem and in geometry. Rigidity and other characteristics of all kinds of hulls are described in terms of such equations. Control problems and fully nonlinear PDEs also arise in engineering, target tracking, pattern recognition, and many other areas. There are many random processes which we want to control, for instance, the performance of a portfolio or the trajectory of a missile. In target tracking it is important to emphasize that the trajectory is usually only observed with certain errors or noises. Thus the first problem is to filter the noise out of the observations. Such problems were first solved by Kalman and Bucy, who constructed and used their filter during the Apollo program. Currently a major problem is to obtain better results in such problems as weather forecasting, which is a very practical application of SPDEs and the theory of filtering and prediction of random processes."

本课题涉及扩散过程和偏微分方程（PDEs）理论中的一些重要的现代课题。它们来源于实际应用，包括随机过程的最优控制，扩散过程的最优滤波，种群遗传学中出现的白噪声驱动的随机偏微分方程（SPDEs），全非线性偏微分方程理论。该项目包括对求解的数值方法的研究。完全非线性偏微分方程出现在最优质量运输问题和几何问题中。用这种方程描述了各种船体的刚度和其他特性。控制问题和完全非线性偏微分方程也出现在工程、目标跟踪、模式识别和许多其他领域。有许多随机过程是我们想要控制的，例如，投资组合的表现或导弹的轨迹。在目标跟踪中，必须强调的是，轨迹通常只有在一定误差或噪声的情况下才能被观察到。因此，第一个问题是将噪声从观测中滤除。这些问题最早是由卡尔曼和布西解决的，他们在阿波罗计划中构建并使用了他们的滤波器。目前的一个主要问题是如何在天气预报等问题上获得更好的结果，这是SPDEs和随机过程滤波预测理论的一个非常实际的应用。”