Various Inverse Problems in Partial Differential Equations and Methods for their Solutions

[9971674] rundell本文研究了偏微分方程的几个反问题。我们特别关注的是唯一性结果，这些结果寻求最小数量的数据，以确定位于某些预定类别中的未知系数。我们也对基于迭代方案的重建方法感兴趣，特别是对利用系数的高导数来进行数据映射的方案。最后，对于某些待定系数问题，我们建议研究以下问题：给定关于数据误差的信息（或假设），即它服从哪种类型的分布以及有效矩是什么，系数可能重建的分布是什么？许多具有物理意义的物体不能直接研究。例子包括人体内部成像，确定固体物体内部的裂缝，以及材料参数，如不可接近物体的导电性。当这些问题被翻译成数学术语时，它们采用偏微分方程的形式，这是数学科学的通用语言。模型中额外的未知数转化为方程中的未知参数，我们试图通过进一步的测量来解决这些问题。在这个建议中，我们从数学的角度来处理这个问题的实际方面：我们感兴趣的问题是什么时候可以做出唯一的决定，以及设计算法来有效地数值恢复未知数。已使用的和有待发展的数学技术适用于若干不同的科学问题。例如：(1)从表面测量中恢复内部物体的位置和形状。这里的应用包括肿瘤（以更高的代谢率为特征）和埋藏的地雷和弹药（以更高的导电性为特征）的测定。(2)利用光谱数据确定太阳内部密度剖面。这很重要，因为它将为我们目前对恒星（如太阳）的组成和我们用来模拟其内部反应的基本物理学的理解提供一个独立的检验。

9971674RundellThis proposal investigates several inverse problems for partial differential equations. Our particular focus is in uniqueness results that seek the minimal amount of data required to determine unknown coefficients lying within certain predetermined classes. We are also interested in reconstruction methods based on iteration schemes and in particular to schemes that utilize higher derivatives of the coefficient to data map. Finally, for certain undetermined coefficient problems we propose to investigate the following: given information (or assumptions) on the error in the data, that is, what type of distribution it obeys and what are the significant moments, what is the distribution of the possible reconstructions of the coefficient?Many objects of physical interest cannot be studied directly. Examples include imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. When these problems are translated into mathematical terms they take the form of partial differential equations, the Lingua Franca of the mathematical sciences. The additional unknowns in the model translate into unknown parameters in the equations and we attempt to solve for these by means of further measurements. In this proposal we deal with the practical aspects of this from a mathematical perspective: We are interested in the question of when a unique determination can be made, as well as designing algorithms for the efficient numerical recovery of the unknowns. The mathematical techniques used and to be developed are applicable to several different scientific problems. Examples are: (1) The recovery of the location and shape of interior objects from surface measurements. Applications here include the determination of tumors (which are characterized by a greater metabolic rate) and buried mines and munitions (characterized by a higher electrical conductivity). (2) The determination of the interior density profile of the sun from spectral data. This is important because it would provide an independent check of our current understanding of both the composition of a star such as the sun and the underlying physics with which we model its internal reactions.