ADVANCE Fellows Award: Inverse Problems for Anisotropic Elastic Media

这个项目的目的是研究有界的、三维的、各向异性弹性介质的动态逆问题的唯一性，也就是说，解决这样一个问题：弹性物体表面的位移-牵引测量是否唯一地决定了内部的密度和弹性特性？特别是，目的是研究该理论是否预测了从表面测量中感知内部特性的任何模糊性，如果是这样，是否有可能表征从表面“隐藏”的弹性介质的特征。PI计划（与Anna Mazzucato合作）通过将边界固定在一阶的微分同态来改变媒介，来呈现这种模糊性，这是对独特性的阻碍。也就是说，我们的计划是证明弹性介质在通过这些微分同态的回拉作用下处于同一轨道时具有相同的狄利克雷-诺伊曼映射。阻碍唯一性的一个后果是，如果在较简单的类别中解决了参数识别问题，则可以部分地解决大类别各向异性弹性介质的参数识别问题。特别是，PI计划将各向同性弹性动力学的唯一性结果扩展到（可能是复合的）各向异性弹性介质的类别。为了应用部分唯一性结果，重要的是开发工具来识别给定的弹性介质是否在某一类弹性介质（例如，各向同性）的轨道上。PI计划通过给出一般各向异性弹性介质在这种作用下的轨道的逐点特征来开始解决这个问题。成像技术帮助医生检测和诊断异常组织。例如，使用超声波的技术最近被开发出来，用于感知生物组织的内部点在表面运动时的反应。关于组织如何运动的信息可以用来识别比周围环境更硬的区域，由于肿瘤通常被包裹在比正常组织更硬的组织中，这种工具可以有重要的医学应用。由于刚度是一种弹性特征，数学家可以应用微分方程领域的方法来研究这个反问题。PI计划与来自主办机构的本科生和研究生合作，在互联网上展示这个反问题的图形结果和指导性文本。该奖项由美国国家科学基金会ADVANCE计划支持。ADVANCE计划的总体任务是通过增加女性在学术科学和工程职业中的代表性和进步来增加女性在科学和工程劳动力中的参与度。

The purpose of this project is to study uniqueness for the dynamic inverse problem for bounded, three-dimensional, anisotropic elastic media, that is, to address the question: do displacement-traction measurements at the surface of an elastic object uniquely determine the density and elastic properties of the interior? In particular, the aim is to study whether the theory predicts any ambiguities in the sensing of the internal properties from surface measurements, and, if so, if it is possible to characterize features of the elastic media that are ``hidden'' from the surface. The PI plans (in joint work with Anna Mazzucato) to present such an ambiguity, an obstruction to uniqueness, by transforming the medium via diffeomorphisms that fix the boundary to first order. That is, the plan is to show that elastic media have the same Dirichlet-to-Neumann map if they lie in the same orbit under the action of pullback via these diffeomorphisms. A consequence of the obstruction to uniqueness is that the parameter identification problem for large classes of anisotropic elastic media may be solved, in part, if it has been solved for simpler classes. In particular, the PI plans to extend uniqueness results for isotropic elastodynamics to classes of (possibly composite) anisotropic elastic media. To apply the partial uniqueness result, it is important to develop tools to identify whether a given elastic medium is in the orbit of a certain class of elastic media (for example, isotropic). The PI plans to begin addressing this problem by giving a pointwise characterization of the orbits of general anisotropic elastic media under this action. Imaging technologies aid physicians in detecting and diagnosing abnormal tissue. Techniques using ultrasound, for example, have been developed recently for sensing how the interior points of biological tissues respond when movement is initiated at the surface. Information about how the tissue moves can be used to identify regions that are stiffer than their surroundings, and since tumors are often encapsulated in tissue which is stiffer than normal, this tool can have important medical applications. Since stiffness is an elastic feature, mathematicians can apply methods from the field of differential equations to contribute to this study of this inverse problem. The PI plans to work with undergraduate and graduate students from the host institution to present graphical results and instructive text on the internet for this inverse problem. This award is supported by the NSF ADVANCE Program. The overall mission of the ADVANCE Program is to increase the participation of women in the scientific and engineering workforce through the increased representation and advancement of women in academic science and engineering careers.