Diffusion Processes and Partial Differential Equations

扩散过程和偏微分方程

• 批准号: 0140405
• 负责人: Nicolai Krylov
• 金额: $ 43.55万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140405&HistoricalAwards=false
• 关键词: Diffusion; Processes; Partial; Differential; Equations

There are three main areas of proposed activity. The first one is developing a technique based on quasiderivatives of solutions of Ito's equations and applying it to processes considered up to the first exit time from domains. The second area is investigating smoothness of solutions of SPDEs arising in filtering problems and the theory of measure--valued processes in order to be able to guarantee certain rates of convergence of approximations to their solutions. The third area is estimating the rate of convergence of numerical approximations for degenerate controlled diffusion processes. The project relates to the investigation of the probabilistic behavior of certain objects. Part one is aimed at better understanding of averaged quantities related to optimal control of random processes arising in all types of applications from finance to aerospace engineering. Part two aims at problems directly related to many practical issues such as image reconstruction or high-performance computing in eliminating "friendly fire". It is also vital in dealing with problems like evolution of bacteria population which may be important in biotechnology. Part three deals with ways of solving practical problems of optimal control of random processes which arise in the applications mentioned above.

拟议活动有三个主要领域。第一个是开发一种技术的基础上quasidifferentiators的解决方案的伊藤方程和应用它的过程中考虑到第一次退出时间域。第二个领域是调查平滑的解决方案的SPDEs所产生的过滤问题和理论的措施-值的过程，以便能够保证一定的速度收敛的近似其解决方案。第三个领域是退化控制扩散过程数值逼近的收敛速度估计。 该项目涉及调查某些物体的概率行为。第一部分旨在更好地理解从金融到航空航天工程的所有类型的应用中产生的随机过程的最佳控制相关的平均量。第二部分主要研究了图像重建、高性能计算等实际问题在排除“误伤”中的应用。它在处理细菌种群进化等问题时也至关重要，这在生物技术中可能很重要。第三部分讨论在上述应用中出现的随机过程最优控制的实际问题的解决方法。