Mathematical Sciences: Inverse Problems in Wave Propagation

数学科学：波传播的反问题

• 批准号: 8902122
• 负责人: Paul Sacks
• 金额: $ 5.24万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1992-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8902122&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Problems; Wave

8902122 Sacks The subject of inverse scattering includes a number of interesting and important scientific problems, arising in the areas of geophysics, quantum mechanics, nondestructive testing, medical imaging and elsewhere. Mathematical formulation of such problems often require the determination of one or more coefficient functions in a partial differential equation (or system of equations), given as data certain limited knowledge about special solutions to the equation. One finds two substantial sources of difficulty associated with most problems of this type, in theory and in application, namely nonlinearity and ill-posedness. There is a need for much more mathematical analysis. The objective of this project is the investigation of the mathematical structure of certain model problems arising in several areas of inverse scattering. As a general rule the approach to a particular nonlinear problem will be via the study of linearized approximations. Such linearized inverse problems are also of independent interest. Specific questions under investigation include the uniqueness of the solution, the continuous dependence of the solution on the given data, the extent of validity of linearization techniques, and how the answers to the previous questions can be used in the design of numerical solution methods.

8902122 Sacks 逆散射主题包括许多有趣且重要的科学问题，这些问题出现在地球物理学、量子力学、无损检测、医学成像和其他领域。 此类问题的数学表述通常需要确定偏微分方程（或方程组）中的一个或多个系数函数，并将有关方程特殊解的某些有限知识作为数据给出。 人们发现，在理论和应用中，大多数此类问题存在两个重要的困难来源，即非线性和不适定性。 需要更多的数学分析。 该项目的目标是研究逆散射几个领域中出现的某些模型问题的数学结构。 作为一般规则，解决特定非线性问题的方法将通过线性近似的研究。 这种线性化反问题也具有独立的意义。 正在研究的具体问题包括解的唯一性、解对给定数据的连续依赖性、线性化技术的有效性程度，以及如何在数值求解方法的设计中使用前面问题的答案。