Reconstruction algorithms for inverse obstacle problems

许多物理参数不能直接研究。示例包括：对身体内部进行成像或定位被掩埋的物体;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。当这些问题被转化为数学术语时，它们采取偏微分方程的形式。然而，由于我们在模型中有额外的未知数，这些在方程中引入了未知参数，我们必须通过进一步的测量来解决这些未知参数。该提案的一个中心主题是什么时候可以做出唯一的决定（所需的最小数据量是多少）以及设计算法以有效地恢复未知数。该建议从数学的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中恢复内部物体的位置、形状和材料特性。在这样的逆问题中，必须始终理解两件事。首先，重建将对数据中的微小变化非常敏感，这是基础物理学所固有的;在数学术语中，这些是包含分析和计算复杂性的高度病态问题。第二，现有的数据总是有误差的。然而，我们可能知道数据误差的模型，例如其均值和方差。该建议寻求一种配方，将使我们能够提供类似的信息的几何形状的障碍-即定量评估的范围内的重建，人们可以预期与给定的数据误差水平。这将使我们能够分配一个概率，一个特定的功能将是可识别的，或者说，物体的体积大于给定的值。该提案有一系列更广泛的影响，其中不仅包括这些反问题所涵盖的科学和工程应用的广度，但有一个重要的培训方面涉及。具体来说，许多问题都有简化的版本，其中所需的实验装置以及一些相应的重建算法都在高年级本科生的能力范围内。这将使更广泛的受众能够了解这些普遍存在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.