Collaboration on Inverse Problems for Holographic Image Datausing KAM Methods

小行星9803498 首席研究员和她的同事乔伊斯·麦克劳克林将 合作解决全息图像数据的逆问题， Kolmogorov-Arnold-Moser（KAM）理论 目标是使用选定的 振动系统模态振型的水平集作为反演的数据 问题.将这些水平集作为数据，建立公式。 的 然后将使用公式来确定系统的物理性质， 例如密度或硬度。 结果将基于微扰 固有频率和振型的结果。 的困难 建立这些结果产生的事实，一个小除数问题， 并且必须同时求解一系列Eikonal方程。 一 由此产生的数学结构的结果将是， 扰动量可以与未扰动量有很大的不同 是介于 麦克劳克林的研究生将专注于开发 使用数据的公式以及这些公式的数值实现 公式。 这项工作的目标是考虑膜状材料，如 生物组织的薄片。 用一个振荡的 力，并假设振荡频率是自然频率， 是膜给出大响应的频率。照明 用两个激光器振动表面，我们看到一个黑暗和光明的线条图案。 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;在某些情况下，它可能更硬或更不硬。 地 在生物学的例子中，增加的刚度可以指示 存在快速分裂的细胞。 在机械的例子中， 硬度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， 找到一个无损检测的刚度变化。 我们的数据就是黑暗 和光线图案。这个问题很难解决，因为它很僵硬 变化可能（但不总是）对系统产生非常大的扰动影响。 格局 当扰动很大时，数学将建立， 当它不是，什么公式将产生刚度变化， 这个特殊的数据集。

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.