Georgia Institute of Technology College of Sciences

I will present some results on the space-time stereo reconstruction of nonlinear sea waves off the Venice coast using a Variational Wave Acquisition Stereo System (VWASS). Energy wave spectra, wave dispersion and nonlinearities are then discussed. The delicate balance of dispersion and nonlinearities may yield the formation of solitons or traveling waves. These are introduced in the context of the Euler equations and the associatedthird order compact Zakharov equation. Traveling waves exist also in the axisymmetric Navier-Stokes equations. Indeed, it will be shown that the NS equations can be reduced to generalized Camassa-Holm equations that support smooth solitons and peakons.

Metamaterials are a new form of structured materials used to control electromagnetic waves through localized resonances. In this talk we introduce a rigorous mathematical framework for controlling localized resonances and predicting exotic behavior inside optical metamaterials. The theory is multiscale in nature and provides a rational basis for designing microstructure using multiphase nonmagnetic materials to create backward wave behavior across prescribed frequency ranges.

Coefficient Inverse Problems (CIPs) are the hardest ones to work with in the field of Inverse Problems. Indeed, they are both nonlinear and ill-posed. Conventional numerical methods for CIPs are based on the least squares minimization. Therefore, these methods suffer from the phenomenon of multiple local minima and ravines. This means in turn that those methods are locally convergent ones. In other words, their convergence is guaranteed only of their starting points of iterations are located in small neighborhoods of true solutions. In the past five years we have developed a new numerical method for CIPs for an important hyperbolic Partial Differential Equation, see, e.g. [1,2] and references cited there. This is a globally convergent method. In other words, there is a rigorous guarantee that this method delivers a good approximation for the exact solution without any advanced knowledge of a small neighborhood of this solution. In simple words, a good first guess is not necessary. This method is verified on many examples of computationally simulated data. In addition, it is verified on experimental data. In this talk we will outline this method and present many numerical examples with the focus on experimental data.REFERENCES [1] L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. [2] A.V. Kuzhuget, L. Beilina and M.V. Klibanov, A. Sullivan, L. Nguyen and M.A. Fiddy, Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28, 095007, 2012.

Images consist of features of varying scales. Thus, multiscale image processing techniques are extremely valuable, especially for medical images. We will discuss multiscale image processing techniques based onvariational methods, specifically (BV, L^2) and (BV, L^1) decompositions. We will discuss the applications to real time denoising, deblurring and image registration.