ADVANCE Fellows Award: Inverse Problems for Anisotropic Elastic Media

这个项目的目的是研究有界、三维、各向异性弹性介质动态反问题的唯一性，即解决这样一个问题：弹性物体表面的位移-牵引力测量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从表面测量的内部性质的传感中有任何模棱两可的地方，如果是的话，是否有可能表征弹性介质对表面“隐藏”的特征。PI计划(与Anna Mazzucato共同工作)通过将边界固定为一阶的微分同胚来变换媒介，从而呈现这样的模糊性，阻碍独特性。也就是说，我们的计划是通过这些微分同态证明，如果弹性介质在拉回作用下位于相同的轨道上，则它们具有相同的Dirichlet-to-Neumann映射。对唯一性的阻碍的一个结果是，如果对于较简单的类别已经解决了大类各向异性弹性介质的参数识别问题，则可以部分地解决该问题。特别是，PI计划将各向同性弹性动力学的唯一性结果推广到各类(可能是复合的)各向异性弹性介质。要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的轨道上(例如，各向同性)。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描述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已经发展起来，用于检测生物组织的内点在表面开始运动时如何反应。有关组织如何移动的信息可以用来识别比周围更僵硬的区域，而且由于肿瘤通常被包裹在比正常组织更坚硬的组织中，因此这种工具可以有重要的医学应用。由于刚度是一种弹性特性，数学家可以应用微分方程组领域的方法来研究这个逆问题。PI计划与主办机构的本科生和研究生合作，在互联网上展示这个逆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先行计划资助。高级方案的总体任务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工作中的参与度。

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.