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.
Monday, March 7, 2016 - 14:00 , Location: Skiles 005 , Prof. Brittany Froese , New Jersey Institute of Technology , Organizer:
The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, Monge-Ampere equations, and non-continuous solutions of the prescribed Gaussian curvature equation.
Monday, February 15, 2016 - 14:05 , Location: Skiles 005 , Professor Hautieng Wu , University of Toronto , Organizer: Haomin Zhou
Explosive technological advances lead to exponential growth of massive data-sets in health-related fields. Of particular important need is an innovative, robust and adaptive acquisition of intrinsic features and metric structure hidden in the massive data-sets. For example, the hidden low dimensional physiological dynamics often expresses itself as atime-varying periodicity and trend in the observed dataset. In this talk, I will discuss how to combine two modern adaptive signal processing techniques, alternating diffusion and concentration of frequency and time(ConceFT), to meet these needs. In addition to the theoreticaljustification, a direct application to the sleep-depth detection problem,ventilator weaning prediction problem and the anesthesia depth problemwill be demonstrated. If time permits, more applications likephotoplethysmography and electrocardiography signal analysis will be discussed.
Monday, January 25, 2016 - 14:00 , Location: Skiles 005 , Predrag Cvitanović , Center for Nonlinear Science, School of Physics, GT , Organizer: Sung Ha Kang
All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. What is the best resolution possible for a given physical system?It turns out that for nonlinear dynamical systems the noise itself is highly nonlinear, with the effective noise different for different regions of system's state space. The best obtainable resolution thus depends on the observed state, the interplay of local stretching/contraction with the smearing due to noise, as well as the memory of its previous states. We show how that is computed, orbit by orbit. But noise also associates to each a finite state space volume, thus helping us by both smoothing out what is deterministically a fractal strange attractor, and restricting the computation to a set of unstable periodic orbits of finite period. By computing the local eigenfunctions of the Fokker-Planck evolution operator, forward operator along stable linearized directions and the adjoint operator along the unstable directions, we determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive noise. The space of all chaotic spatiotemporal states is infinite, but noise kindly coarse-grains it into a finite set of resolvable states.(This is work by Jeffrey M. Heninger, Domenico Lippolis,and Predrag Cvitanović,arXiv:0902.4269
, arXiv:1206.5506 and arXiv:1507.00462 )
Monday, December 7, 2015 - 14:05 , Location: Skiles 005 , Professor Jun Zhang , Courant Institute , Organizer: Martin Short
Thermal convection is ubiquitous in nature. It spans from a small cup of tea to the internal dynamics of the earth. In this talk, I will discuss a few experiments where boundaries to the fluid play surprising roles in changing the behaviors of a classical Rayleigh- Bénard convection system. In one, mobile boundaries lead to regular large-scale oscillations that involve the entire system. This could be related to the continental kinetics on earth over the past two billion years, as super-continents formed and broke apart in cyclic fashion. In another experiment, we found that seemingly impeding partitions in thermal convection can boost the overall heat transport by several folds, once the partitions are properly arranged, thanks to an unexpected symmetry-breaking bifurcation.