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

首席研究员和她的同事尤利娅·卡佩希娜将合作解决全息图像数据的反问题，使用Kolmogorov-Arnold-Moser(KAM)理论。目标是使用选定的振动系统振型的水平集作为反问题的数据。以这些水平集为数据，建立公式。然后，这些公式将被用来确定系统的物理属性，如密度或刚度。结果将基于对固有频率和振型的摄动结果。建立这些结果的困难源于这样一个事实，即必须同时求解一个小因子问题和一系列Eikonal方程。所得到的数学结构的结果将是扰动量可能与未扰动量有很大的不同。麦克劳克林的研究生将专注于开发使用这些数据的公式，并对这些公式进行数值实现。这项工作的目标是考虑膜状材料，如生物组织的薄片。用振荡力激励这个膜，假设振动的频率是一个自然频率，也就是膜产生大响应的频率。用两个激光照射振动的表面，我们看到了一种暗淡的线条图案。每一条直线都是振动表面的一个水平集。现在假设膜是不均匀的；它可能在某些地方变得更硬或更不硬。[在生物学的例子中，僵硬的增加可以表明存在快速分裂的细胞。在一个机械的例子中，刚性降低可能表明材料的劣化。]目标是在不改变膜的情况下确定刚度变化，即找到一种无损测试刚度变化的方法。我们的数据是深浅相间的线条图案。这个问题是困难的，因为刚度的变化可能会对图案产生非常大的扰动影响(但并不总是)。数学将确定什么时候扰动很大，什么时候不大，以及从这个特定的数据集得出刚度变化的公式是什么。

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.