Inverse Problems for Anisotropic Media

本文主要研究各向异性介质的反问题，特别是稳定性、全局唯一性和边界确定问题。在第一个项目中（与a . Sa Barreto联合），PI将研究光滑黎曼度规g（定义在有界的n维域上）在与拉普拉斯-贝尔特拉米算子的波动方程相关的Dirichlet-to-Neumann映射上的连续依赖性，直到通过固定边界的微分态射回调。第二个项目是关于某一类各向异性介质的全局唯一性。这里开发的技术将应用于具有初始应力的弹性动力学逆问题。第三个项目的目的是描述具有初始应力的弹性动力学算子的系数在边界处由Dirichlet-to-Neumann映射确定为无限阶的条件。在不同的环境中出现，逆问题作为数学技术应用于实际问题的潜力的引人注目的例子脱颖而出。逆问题涉及描述内部性质，如飞机机翼的导电性，地下矿藏的弹性，或人体组织的密度，只给出外部测量。例如，对于通过弹性动力学逆问题研究的物体，已知在某些情况下，物体的密度和弹性特性是由在表面进行的测量唯一确定的。也就是说，如果两个具有相同大小、形状和方向的物体对仅在表面进行的某些测试有相同的反应，那么可以保证它们实际上具有相同的密度和弹性特性。在上面提到的第二个和第三个项目中，PI将考虑具有初始应力的弹性物体。过去的事件可能会在物体中建立一个初始应力，例如，通过焊接或地震断层活动。然而，在任何实际测量中都存在一些固有的误差，因此上面提到的第一个项目的目的是表明，为了从近似的表面测量中任意准确地重建物体的属性，必须考虑某些模糊性。

Liz Rachele (DMS-9801664)This proposal concerns inverse problems for anisotropic media, in particular, questions on stability, global uniqueness, and boundary determination. In the first project (joint with A. Sa Barreto) the PI will study the continuous dependence of a smooth Riemannian metric g (defined on a bounded, n-dimensional domain) on the Dirichlet-to-Neumann map associated with the wave equation for the Laplace-Beltrami operator, up to the pullback by a diffeo-morphism that fixes the boundary. The second project is concerned with global uniquenessfor a certain class of anisotropic media. Techniques developed here will be applied to an inverse problem for elastodynamics with initial stress. The aim of the third project is to describe conditions under which the coefficients of the operator for elastodynamics with initial stress are determined to infinite order at the boundary by the Dirichlet-to-Neumann map. Arising in diverse settings, inverse problems stand out as compelling examples of the potential for application of mathematical techniques to practical problems. Inverse problems involve describing internal properties, such as the conductivity of an airplane wing, the elasticity of subterranean ore deposits, or the density of human tissue, given only measurements made externally. For objects studied via an inverse problem in elasto-dynamics, for example, it is known in some cases that the density and elastic properties of the objects are uniquely determined by measurements made at the surface. That is, if twoobjects with the same size, shape, and orientation respond in the same way to certain tests made only at the surface, then it is guaranteed that they do, in fact, have the same density and elastic properties throughout. In the second and third projects mentioned above the PI will consider elastic objects with initial stress. Events in the past may have built up an initial stress in the object, for example, through welding or seismic fault activity. Some error is inherent in any real measurements, though, and so the aim of the first project mentioned above is to show that only certain ambiguities must be taken into account in order to reconstruct properties of the object arbitrarily accurately from approximate surface measurements.