Mathematical Sciences: Well-Posed Inverse Problems

数学科学：适定反问题

• 批准号: 9305882
• 负责人: James Ralston
• 金额: $ 14.32万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-05-15 至 1996-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9305882&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Well; Posed; Inverse

The object of this work, the study of inverse problems in mathematical analysis, is one of recovering physical properties of an object or phenomenon from data obtained by remote observations. The problems are well-posed in the sense that the data sets are not so burdensome as to rule out any solutions, but sufficient for the reconstruction. The problem of determining a quantum mechanical potential from its scattering amplitude goes back to the beginning of quantum mechanics. But little attention is given to finding data sets for which the problem is well-posed in the sense that the mapping from the potential to the scattering data is continuously invertible. One likely candidate for such a data set is backscattering data, the subject of this research. Much successful work has been accomplished on the question in three dimensions, though mainly confined to small potentials. The present work seeks to remove this assumption and look for more general results, possibly giving up some degree of well-posedness. Work will continue on two-dimensional backscattering as well, where a type of singularity in a parameter makes backscattering a much more untractable problem, even assuming compact support of the potential.Inverse problems in the context of this project are formulated in terms of classical equations, such as the Schrodinger equation. These equations may represent wave phenomena involving magnetic and electric potentials, acoustics and other type of waves in the presence of obstacles. Although much of the research focuses on the mathematical analysis of the inverse mapping, its connections with the physical world are easily traced.

这项工作的目的是数学分析中的反问题的研究，是从远程观察获得的数据中恢复对象或现象的物理特性之一。 这些问题是有充分意义的，因为数据集并非如此繁重，以至于排除任何解决方案，但足以进行重建。从其散射幅度确定量子力势的问题回到了量子力学的开始。 但是，很少有人注意找到问题的数据集，因为从电势到散射数据的映射是不断可逆的。 此类数据集的一个可能候选人是反向散射数据，这是这项研究的主题。 在三个维度上的问题上已经完成了许多成功的工作，尽管主要限于小潜力。 目前的工作旨在消除这一假设并寻找更一般的结果，可能会放弃一定程度的适应性。 工作也将继续进行二维反向散射，在这种情况下，参数中的一种奇异性使反向散射变得更加不可吸引，即使假设对潜在的支持进行了紧凑的支持。在该项目的背景下，以经典方程式（例如Schrodinger方程式）提出了反向问题。这些方程可能代表涉及磁性和电势，声学和其他类型波的波浪现象。 尽管许多研究都集中在反映射的数学分析上，但它与物理世界的联系很容易追踪。