Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Many real-world problems reduce to optimization problems that are solved by iterative methods. In this talk, I focus on recently developed efficient algorithms for solving large-scale optimization problems that arises in medical imaging and image processing. In the first part of my talk, I will introduce the Bregman Operator Splitting with Variable Stepsize (BOSVS) algorithm for solving nonsmooth inverse problems. The proposed algorithm is designed to handle applications where the matrix in the fidelity term is large, dense, and ill-conditioned. Numerical results are provided using test problems from parallel magnetic resonance imaging. In the second part, I will focus on the Euler's Elastica-based model which is non-smooth and non-convex, and involves high-order derivatives. I introduce two efficient alternating minimization methods based on operator splitting and alternating direction method of multipliers, where subproblems can be solved efficiently by Fourier transforms and shrinkage operators. I present the analytical properties of each algorithm, as well as several numerical experiments on image inpainting problems, including comparison with some existing state-of-art methods to show the efficiency and the effectiveness of the proposed methods.

An adaptive hybrid level set moment-of-fluid method is developed to study the material solidification of static and dynamic multiphase systems. The main focus is on the solidification of water droplets, which may undergo normal or supercooled freezing. We model the different regimes of freezing such as supercooling, nucleation, recalescence, isothermal freezing and solid cooling accordingly to capture physical dynamics during impact and solidification of water droplets onto solid surfaces. The numerical simulations are validated by comparison to analytical results and experimental observations. The present simulations demonstrate the ability of the method to capture sharp solidification front, handle contact line dynamics, and the simultaneous impact, merging and freezing of a drop. Parameter studies have been conducted, which show the influence of the Stefan number on the regularity of the shape of frozen droplets. Also, it is shown that impacting droplets with different sizes create ice shapes which are uniform near the impact point and become dissimilar away from it. In addition, surface wettability determines whether droplets freeze upon impact or bounce away.

A nonlinear filtering problem can be classified as a stochastic Bayesian optimization problem of identifying the state of a stochastic dynamical system based on noisy observations of the system. Well known numerical simulation methods include unscented Kalman filters and particle filters. In this talk, we consider  a class of efficient numerical methods based on  forward backward stochastic differential equations. The backward SDEs for nonlinear filtering problems are similar to the Fokker-Planck equations for SDEs.  We will describe the process of deriving such backward SDEs as well as high order numerical algorithms to solve them, which in turn solve nonlinear filtering problems.

Language change is a complex social phenomenon, revealing pathways of communication and sociocultural influence. But while language change has long been a topic of study in sociolinguistics, traditional linguistic research methods rely on circumstantial evidence, estimating the direction of change from differences between older and younger speakers. In this research, we use a data set of several million Twitter users to track language changes in progress. First, we show that language change can be viewed as a form of social influence: we observe complex contagion for ``netspeak'' abbreviations (e.g., lol) and phonetic spellings, but not for older dialect markers from spoken language. Next, we test whether specific types of social network connections are more influential than others, using a parametric Hawkes process model. We find that tie strength plays an important role: densely embedded social ties are significantly better conduits of linguistic influence. Geographic locality appears to play a more limited role: we find relatively little evidence to support the hypothesis that individuals are more influenced by geographically local social ties, even in the usage of geographical dialect markers.