Collaboration on Inverse Problems for Holographic Image Datausing KAM Methods

首席研究员和她的同事Joyce McLaughlin将利用Kolmogorov-Arnold-Moser （KAM）理论合作解决全息图像数据的逆问题。目标是使用振动系统模态振型的选定水平集作为反问题的数据。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，如密度或刚度。结果将基于固有频率和模态振型的扰动结果。建立这些结果的困难是由于一个小因子问题和一系列Eikonal方程必须同时求解。由此产生的数学结构的一个结果是，摄动量可以与未摄动量有很大的不同。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类膜材料，如生物组织的薄片。用一个振荡力激发这个膜，假设振荡频率是固有频率，也就是说，在这个频率上膜会产生很大的响应。用两束激光照亮振动表面，我们会看到一条暗线和一条亮线。每条线是振动面的一个水平集。现在假设膜是非均匀的；在某些地方可能会更硬或不那么硬。在生物学的例子中，增加的硬度可以表明存在快速分裂的细胞。在一个机械例子中，刚度的降低可以表明材料的劣化。目标是在不改变膜的情况下确定刚度变化，即找到一种无损检测刚度变化的方法。我们的数据是暗线和亮线模式。这个问题是困难的，因为刚度的变化可以（但并不总是）有一个非常大的扰动影响的模式。数学将建立什么时候扰动大，什么时候扰动小，以及什么公式将从这个特定的数据集产生刚度变化。

9803498 Karpeshina The principal investigator and her colleague Joyce McLaughlin 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 be cause 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.