Monday, October 24, 2016 - 14:05 , Location: Skiles 005 , Prof. Lars Ruthotto , Emory University Math/CS , Organizer: Martin Short
Image registration is an essential task in almost all areas involving imaging techniques. The goal of image registration is to find geometrical correspondences between two or more images. Image registration is commonly phrased as a variational problem that is known to be ill-posed and thus regularization is commonly used to ensure existence of solutions and/or introduce prior knowledge about the application in mind. Many relevant applications, e.g., in biomedical imaging, require that plausible transformations are diffeomorphic, i.e., smooth mappings with a smooth inverse. This talk will present and compare two modeling strategies and numerical approaches to diffeomorphic image registration. First, we will discuss regularization approaches based on nonlinear elasticity. Second, we will phrase image registration as an optimal control problem involving hyperbolic PDEs which is similar to the popular framework of Large Deformation Diffeomorphic Metric Mapping (LDDMM). Finally, we will consider computational aspects and present numerical results for real-life medical imaging problems.
Monday, October 17, 2016 - 14:05 , Location: Skiles 005 , Prof. Yanzhao Cao , Auburn University Mathematics , Organizer: Martin Short
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.
Monday, October 3, 2016 - 14:00 , Location: Skiles 005 , Prof. Qin Li , UW-Madison , firstname.lastname@example.org , Organizer: Molei Tao
Many kinetic equations have the corresponding fluid limits. In the zero limit of the Knudsen number, one derives the Euler equation out of the Boltzmann equation and the heat equation out of the radiative transfer equation. While there are good numerical solvers for both kinetic and fluid equations, it is not quite well-understood when the two regimes co-exist. In this talk, we model the layer between the fluid and the kinetic using a half-space equation, study the well-posedness, design a numerical solver, and utilize it to couple the two sets of equations that govern separate domains. It is a joint work with Jianfeng Lu and Weiran Sun.
Monday, September 12, 2016 - 14:05 , Location: Skiles 005 , Prof. Jacob Eisenstein , GA Tech School of Interactive Computing , Organizer: Martin Short
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.
Monday, August 8, 2016 - 14:00 , Location: Skiles 006 , Prof. Yunho Kim , UNIST, Korea , Organizer: Sung Ha Kang
Inspired by the usefulness of difference of convex functions in some problems, e.g. sparse representations, we use such an idea of difference of convex functions to propose a method of finding an eigenfunction of a self-adjointoperator. In a matrix setting, this method always finds an eigenvector of a symmetric matrix corresponding to the smallest eigenvalue without solving Ax=b. In fact, such a matrix A is allowed to be singular, as well. We can apply the same setting to a generalized eigenvalue problem. We will discuss its convergence as well.
Wednesday, June 22, 2016 - 11:00 , Location: Skiles 006 , Dr. Ha Quang, Minh , Istituto Italiano di Tecnologia (Italy) , Organizer: Sung Ha Kang
Symmetric positive definite (SPD) matrices play important roles in numerous areas of mathematics, statistics, and their applications in machine learning, optimization, computer vision, and related fields. Among the most important topics in the study of SPD matrices are the distances between them that can properly capture the geometry of the set of SPD matrices. Two of the most widely used distances are the affine-invariant distance and the Log-Euclidean distance, which are geodesic distances corresponding to two different Riemannian metrics on this set. In this talk, we present our recently developed concept of Log-Hilbert-Schmidt (Log-HS) distance between positive definite Hilbert-Schmidt operators on a Hilbert space.This is the generalization of the Log-Euclidean distance between SPD matrices to the infinite-dimensional setting. In the case of reproducing kernel Hilbert space (RKHS) covariance operators, we obtain closed form formulas for the Log-HS distance, expressed via Gram matrices. As a practical example, we demonstrate an application of the Log-HS distance to the problem of image classification in computer vision.
Monday, April 11, 2016 - 14:05 , Location: Skiles 005 , Byungmoon Kim , Adobe Research , Organizer: Yingjie Liu
This talk will tell the story on using simulation for painting. I will tell a few of projects that had simulation and painting involved. One is iPad-based ultra-low-cost real time simulation of old photography process to compute effects that modern day users may find interesting. The other is more full-blown fluid simulation for painting using highest-end GPU. Even with massive processing power of GPU, real time high fidelity painting simulation is hard since computation budget is limited. Basically we should deal with large errors. It may sound odd if someone says that very low-accuracy simulation is interesting - but this is very true. In particular, we tried to pull most pressure effect out from about 10 Jacobi iterations that we could afford. I would like to share my experience on improving fixed number of fixed point iterations.
Friday, April 8, 2016 - 14:00 , Location: Skiles 005 , Prof. Ming-Jun Lai , Department of Mathematics, University of Georgia , Organizer: Sung Ha Kang
Bivariate splines are smooth piecewise polynomial functions defined on a triangulation of arbitrary polygon. They are extremely useful for numerical solution of PDE, scattered data interpolation and fitting, statistical data analysis, and etc.. In this talk, I shall explain its new application to a biological study. Mainly, I will explain how to use them to numerically solve a type of nonlinear diffusive time dependent PDE which arise from a biological study on the density of species over a region of interest. I apply our spline solution to simulate a real life study on malaria diseases in Bandiagara, Mali. Our numerical result show some similarity with the pattern from the biological study in2013 in a blind testing. In addition, I shall explain how to use bivariate splines to numerically solve several systems of diffusive PDEs: e.g. predator-prey type, resource competing type and other type systems.
Monday, April 4, 2016 - 14:00 , Location: Skiles 005 , Wuchen Li , Georgia Tech Mathematics , Organizer: Martin Short
Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them.In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory.In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem.In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.
Monday, March 28, 2016 - 14:05 , Location: Skiles 005 , Zhilin Li , North Carolina State University , Organizer:
In this talk, I will introduce the Immersed Finite Element Methods (IFEM) for one and two dimensional elliptic interface problems based on Cartesian triangulations. The key is to modify the basis functions so that the homogeneous jump conditions are satisfied in the presence of discontinuity in the coefficients. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. For non-homogeneous jump conditions, we have developed a new strategy to transform the original interface problem to a new one with homogeneous jump conditions using the level set function. If time permits, I will also explain some recent progress in this direction including the augmented IFEM for piecewise constant coefficient, and a SVD free version of the method.