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

9971674Rundell这个提议研究了偏微分方程的几个反问题。 我们特别关注的是在唯一性的结果，寻求所需的数据量最小，以确定未知系数躺在某些预定的类。我们也有兴趣在重建方法的基础上迭代计划，特别是计划，利用更高的导数的系数数据映射。最后，对于某些不确定的系数问题，我们建议研究以下内容：给定数据误差的信息（或假设），即它服从什么类型的分布以及重要时刻是什么，可能的分布是什么系数的重建？许多物理感兴趣的对象不能直接研究。例子包括身体内部的成像，固体物体内裂缝的确定，以及材料参数，如不可接近物体的导电性。当这些问题被转化为数学术语时，它们采取偏微分方程的形式，数学科学的语言弗兰卡。模型中的额外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：我们感兴趣的问题时，一个独特的决定，以及设计算法的有效数值恢复的未知数。所使用的和将要发展的数学技术适用于几个不同的科学问题。实例有：（1）从表面测量恢复内部物体的位置和形状。这里的应用包括确定肿瘤（其特征在于更高的代谢率）和埋在地下的地雷和弹药（其特征在于更高的电导率）。(2)从光谱数据确定太阳内部密度分布。这一点很重要，因为它将为我们目前对太阳等星星的组成以及我们模拟其内部反应的基础物理学的理解提供独立的检查。

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.