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.

Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics. This is joint work with Olivier Zindy.