Collaboration on Inverse Problems Using Holographic Image Data; Using RAM Theory

首席研究员和她的同事Yulia Karpeshina将合作使用Kolmogorov-Arnold-Moser（KAM）理论解决全息图像数据的逆问题。 我们的目标是使用选定的水平集的振动系统的模态作为反问题的数据。将这些水平集作为数据，建立公式。 然后，这些公式将用于确定系统的物理特性，例如密度或刚度。 结果将基于固有频率和振型的扰动结果。 建立这些结果的困难来自于这样一个事实，即一个小的除数问题和Eikonal方程序列必须同时解决。 由此产生的数学结构的结果将是，扰动量可以与未扰动量强烈不同。 McLaughlin的研究生将专注于开发使用这些数据的公式以及这些公式的数值实现。这项工作的目标是考虑膜状材料，例如生物组织的薄片。 用振荡力激励该膜，并假设振荡频率是固有频率，即膜给出大响应的频率。用两束激光照射振动的表面，我们看到一条明暗相间的线条图案。 每条线都是振动表面的一个水平集。 现在假设膜是不均匀的;它在某些地方可能更硬或更不硬。 [In在生物学的例子中，增加的硬度可以指示快速分裂的细胞的存在。 在机械示例中，刚度降低可以指示材料的劣化。目标是在不改变膜的情况下确定刚度变化，即找到刚度变化的无损检测方法。 我们的数据是明暗线模式。这个问题是困难的，因为刚度变化可能（但不总是）对图案有非常大的扰动影响。 数学将确定何时扰动较大，何时扰动较小，以及哪些公式将从该特定数据集产生刚度变化。

The principal investigator and her colleague Yulia Karpeshina will collaborate to solve inverse problems for holographic image data using Kolmogorov-Arnold-Moser (KAM) theory. The goal is to use selected level sets of mode shapes of vibrating systems as data for the inverse problem. With these level sets as data, formulas will be established. The formulas will then be used to determine physical properties of the system, such as density or stiffness. The results will be based on perturbation results for the natural frequencies and the mode shapes. The difficulty in establishing these results arises from the fact that a small divisor problem and a sequence of Eikonal equations must be solved simultaneously. A consequence of the resultant mathematical structure will be that the perturbed quantities can be strongly different from the unperturbed quantities. McLaughlin's graduate student will concentrate on developing formulas to use the data and on numerical implementation of those formulas.The goal with this work is to consider membrane like materials, such as a thin slice of biological tissue. Excite this membrane with an oscillating force and suppose the frequency of oscillation is a natural frequency, that is, a frequency where the membrane gives a large response. Illuminating the vibrating surface with two lasers we see a dark and light line pattern. Each line is a level set of the vibrating surface. Now assume that the membrane is nonhomogeneous; it could be more stiff or less stiff in some places. [In the biological example, increased stiffness can indicate the presence of rapidly dividing cells. In a mechanical example, decreased stiffness can indicate deterioration of the material.] The goal is to determine the stiffness variations without altering the membrane, that is, to find a nondestructive test for the stiffness variations. Our data is the dark and light line pattern. The problem is difficult because the stiffness variations can have (but not always) a very large perturbative effect on the pattern. The mathematics will establish when the perturbation is large, when it is not, and what formulas will yield the stiffness variations from this particular data set.