ADVANCE Fellows Award: Inverse Problems for Anisotropic Elastic Media

本项目的目的是研究有界三维各向异性弹性介质动力学反问题的唯一性，即解决这样一个问题：弹性物体表面的位移-牵引测量值是否唯一地决定了内部的密度和弹性性质？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属性的任何模糊性，如果是这样，是否有可能表征从表面“隐藏”的弹性介质特征。 PI计划（与安娜马祖卡托联合工作）通过将边界固定为一阶的同构变换介质来呈现这种模糊性，这是唯一性的障碍。 也就是说，我们的计划是要证明，如果弹性介质通过这些超同态在拉回的作用下位于相同的轨道，那么它们具有相同的Dirichlet-to-Neumann映射。 唯一性障碍的后果是，大类各向异性弹性介质的参数识别问题可以部分解决，如果它已经解决了较简单的类。 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）各向异性弹性介质类。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类弹性介质（例如，各向同性）的轨道中。PI计划开始通过给出一般各向异性弹性介质在此作用下的轨道的逐点表征来解决这个问题。成像技术帮助医生检测和诊断异常组织。例如，最近已经开发了使用超声波的技术，用于感测当在表面处开始运动时生物组织的内部点如何响应。关于组织如何移动的信息可以用来识别比周围环境更硬的区域，并且由于肿瘤通常被封装在比正常更硬的组织中，因此该工具可以具有重要的医学应用。由于刚度是弹性特征，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，在互联网上为这个逆问题提供图形结果和指导性文本。 该奖项由NSF ADVANCE计划支持。高级方案的总体使命是通过增加妇女在学术科学和工程职业中的代表性和地位，增加妇女在科学和工程劳动力中的参与。

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.