Reconstruction algorithms for inverse obstacle problems

许多具有物理意义的参数不能直接研究。例如：成像身体内部或定位埋藏的物体；确定固体物体内裂缝的位置和大小；重建材料参数，如内部区域的导电性。当这些问题转化为数学术语时，它们就会以偏微分方程式的形式出现。然而，由于我们在模型中有额外的未知数，这些未知数在方程中引入了未知参数，我们必须通过进一步的测量来另外解决这些参数。该提案的一个中心主题是何时可以作出独特的确定(所需的最低数据量是多少)，以及如何设计有效地以数字方式恢复未知数的算法。这项建议从数学角度考虑了这一点的实际和计算方面。解决的具体问题包括从表面测量恢复内部物体的位置、形状和材料属性。在这样的反问题中，必须始终理解两件事。首先，重建将对数据的微小变化极其敏感，这是基本物理学固有的；用数学术语来说，这些是包含分析和计算复杂性的高度病态问题。其次，现有的数据总是容易出错。然而，我们可以知道数据误差的模型，例如其均值和方差。这项提议寻求一种公式，使我们能够提供有关障碍几何形状的类似信息--即对人们在给定数据误差水平下可能预期的重建范围进行定量评估。这将允许我们分配一个概率，即一个特定的特征将是可识别的，或者说，物体的体积大于给定值。该提议具有一系列更广泛的影响。这些影响不仅包括这些反问题所涵盖的科学和工程应用的广度，而且还涉及到一个重要的培训方面。具体地说，许多问题都有简化的版本，其中所需的实验设备以及一些相应的重建算法都是高级本科生可以接触到的。这将使更广泛的受众了解这些普遍但复杂的问题的挑战和可能的解决方案。

Many parameters of physical interest cannot be studied directly. Examples include: imaging the interior of the body or locating buried objects; determining the location and size of cracks within solid objects; reconstructing material parameters such as the conductivity of interior regions. When these problems are translated into mathematical terms they take the form of partial differential equations. However, since we have additional unknowns in the model, these introduce unknown parameters in the equations that we must additionally resolve by means of further measurements. A central theme of the proposal is the question of when a unique determination can be made (what is the minimal amount of data needed) as well as the design of algorithms for the efficient numerical recovery of the unknowns. This proposal considers the practical and computational aspects of this from a mathematical perspective. Specific problems addressed include the recovery of the location, shape, and material properties of interior objects from surface measurements. In such inverse problems two things must always be understood. First, the reconstructions will be extremely sensitive to small changes in the data, this is inherent in the underlying physics; in mathematical terms these are highly ill-conditioned problems containing both analytical and computational complexity. Second, the available data is always subject to error. However, we may know a model for the data error such as, for example, its mean and variance. This proposal seeks a formulation that will allow us to provide similar information on the geometry of the obstacle - namely a quantitative assessment of the ranges of reconstructions one could expect with a given level of data error. This would allow us to assign a probability that a particular feature would be identifiable or that, say, the volume of the object is greater than a given value.The proposal has a range of broader impacts.These include not only the breadth of applications to science and engineering covered by these inverse problems, but there is an important training aspect involved. Specifically, many of the problems have simplified versions where both the experimental apparatus needed as well as some of the corresponding reconstruction algorithms are within reach of advanced undergraduates. This will enable a wider audience to gain an understanding of both the challenges and possible solutions to these ubiquitous but complex problems.