Georgia Institute of Technology College of Sciences

We present a multiscale approach for identifying objects submerged in ocean beds by solving inverse problems in high frequency seafloor acoustics. The setting is based on Sound Navigation And Ranging (SONAR) imaging used in scientific, commercial, and military applications. The forward model incorporates simulations, by solving Helmholtz equations, on a wide range of spatial scales in the seafloor geometry. This allows for detailed recovery of seafloor parameters including the material type. Simulated backscattered data is generated using microlocal analysis techniques. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor parametrizations.

A fundamental result in Harmonic Analysis states that many functions defined over the interval [-\pi,\pi] can be decomposed into a Fourier series, that is, decomposed as sums of sines and cosines with integer frequencies. This allows one to describe very complicated functions in a simple way, and therefore provides with a strong tool to study the properties of different families of functions.However, the above decomposition does not hold -- or holds but is not efficient enough-- if the functions are no longer defined over an interval,( e.g. if a function is defined over a union of two disjoint intervals). We will discuss the question of whether similar decompositions are possible also in such cases, with the frequencies of the sines and cosines possibly being no longer integers.

Abstract: Certain materials form geometric structures called "grains," which means that one has distinct volumes filled with the same semi-solid material but not mixing. This can happen with semi-molten copper and something like this can also happen with liquid crystals (which are used in some calculator display screens). People who try to analyze such systems tend to be interested in the motion of the boundaries between grains (which are often modeled by mean curvature flow) and the motions of the exterior surfaces of grains (which are often modeled by surface diffusion flow). Surfaces of constant mean curvature are stationary for both flows and provide stationary or equilibrium configurations. The surfaces of constant mean curvature which are axially symmetric have been classified. Grain boundaries are not usually axially symmetric, but I will describe a model situation in which they are and one can study the resulting equilibria. I will give a very informal introduction to the flow problems mentioned above (about which I know very little) and then go over the classification of axially symmetric constant mean curvature surfaces (about which I know rather more) and some reasonable questions one can ask (and hopefully answer) about such problems.

Abstract: If P(z) is a polynomial, then log|P(z)| is a potential. We discuss some facets of this observation, and some gems in classical potential theory. A special topics course on potential theory will be offered in the fall.