Georgia Institute of Technology College of Sciences

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.

Given F, finite set of square matrices of dimension n, it is possible to define the Joint Spectral Radius or simply JSR as a generalization of the well known spectral radius of a matrix. The JSR evaluation proves to be useful for instance in the analysis of the asymptotic behavior of solutions of linear difference equations with variable coefficients, in the construction of compactly supported wavelets of and many others contexts. This quantity proves, however, to be hard to compute in general. Gripenberg in 1996 proposed an algorithm for the computation of lower and upper bounds to the JSR based on a four member inequality and a branch and bound technique. In this talk we describe a generalization of Gripenberg's method based on ellipsoidal norms that achieve a tighter upper bound, speeding up the approximation of the JSR. We show the performance of this new algorithm compared with Gripenberg's one. This talk is based on joint work with V.Y.Protasov.

In this talk, I will present two examples of the application of wavelet analysis to the understanding of mild Traumatic Brain Injury (mTBI). First, the talk will focus on how wavelet-based features can be used to define important characteristics of blast-related acceleration and pressure signatures, and how these can be used to drive a Naïve Bayes classifier using wavelet packets. Later, some recent progress on the use of wavelets for data-driven clustering of brain regions and the characterization of functional network dynamics related to mTBI will be discussed. In particular, because neurological time series such as the ones obtained from an fMRI scan belong to the class of long term memory processes (also referred to as 1/f-like processes), the use of wavelet analysis guarantees optimal theoretical whitening properties and leads to better clusters compared to classical seed-based approaches.