Mathematical Sciences: Inverse Problems in Wave Propagation

8902122逆散射的主题包括一些有趣和重要的科学问题，这些问题产生于地球物理、量子力学、无损检测、医学成像和其他方面。这类问题的数学表述通常需要确定偏微分方程式(或方程组)中的一个或多个系数函数，作为数据给定关于该方程特解的某些有限知识。在理论和应用上，人们发现与大多数这类问题相关的两个实质性的困难来源，即非线性和不适定。有必要进行更多的数学分析。这个项目的目标是研究在逆散射的几个领域中出现的某些模型问题的数学结构。一般情况下，解决特定非线性问题的方法是通过研究线性化近似。这种线性化的反问题也具有独立的意义。研究中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，以及如何将上述问题的答案用于数值解方法的设计。

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.