Mathematical Sciences: Inverse Problems in Wave Propagation

逆散射的主题包括许多有趣和重要的科学问题，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。这类问题的数学公式通常需要在偏微分方程（或方程组）中确定一个或多个系数函数，并给出关于该方程特殊解的某些有限知识作为数据。在理论和应用中，人们发现与大多数这类问题相关的困难有两个主要来源，即非线性和不适定性。有必要进行更多的数学分析。本项目的目的是研究在几个逆散射领域中出现的某些模型问题的数学结构。一般来说，解决特定非线性问题的方法是通过研究线性化近似。这种线性化的逆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，以及如何在数值解方法的设计中使用前面问题的答案。

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.