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.
