University of Georgia

Mathematics Department

Applied Mathematics Seminar

All talks are given in Room 326, Boyd Graduate Studies Building on Fridays, 12:20pm--1:10pm, unless otherwise noted. There will be some pizzas and drinks available before the seminar.

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  • Talk Date: 20070202

    Multivariate splines for numerical solution of PDE

    Ming-Jun Lai

    Abstract
    Multivariate splines are smooth piecewise polynomial functions defined over tria ngulated domains. I will^M explain how to use them for numerical solution of some^M linear and nonlinear PDE without constructing finite element method or locally s upported basis functions. ^M A special iteration method for the associted linear systems will be introduced t o show that the method converges. Some numerical examples in MATLAB for Poisson^ M More about speaker, check here.

  • Talk Date: 20070209

    Mathematical Biology and Ecological Systems

    Caner Kazanci

    Abstract
    In this talk, I present some of the mathematical problems arising in biological sciences in general. I will then focus on ecological applications, and introduce Ecological Network Analysis, a graph theoretical approach to study energy or material flow in ecosystems. I will discuss a new stochastic method that we developed, called Particle Tracking Algorithm. Similar to a microscope, this method enables us to observe each and individual energy (or mass) packets flow in the network; and we can simultaneously label and track all of their movements. Unlike agent or individual based models (ABM, IBM) where particles move according to a pre-defined algorithm, particle flow occurs ‘naturally’ (compatible with master equation) without human intervention. This brings the possibility of investigating all predefined ecological network properties, such as cycling index, residence time, dominance of indirect effects, and evaluate how well these algebraic definitions reflect their meaning. More about speaker, check here.

  • Talk Date: 20070216

    The Replicator Equations from Evolutionary Game Theory

    Malcolm Adams

    Abstract
    The replicator equations are a system of ordinary differential equations that describe the evolution of n competing populations in terms of the relative fitness of the various components of the population. I will give a brief derivation of these equations and discuss some properties and classical examples. If time permits, I will also describe recent results of myself and Sornborger on the long term behavior of solutions for a certain class of these equations. More about speaker, check here.

  • Talk Date: February 23

    New challenges of Information Theory.

    Alexander Petukhov

    Abstract
    Mathematical aspects of fundamental problems of Information Theory will be discussed, which became hot topics during last two years. Many leading mathematicians, statisticians, and computer scientists switched to these problems. A very incomplete list of them includes D. Donoho, T. Tao, R. Coifman, I. Daubechies, V. Tarokh, E. Candes. The topics include Compression (and Compessed Sensing as a special case), Transmission in Noisy Channels, and Cryptography for real-valued (or, more generally, metric space) data. Instant applications of these problems are digital broadcasting, mobile telephony, the storage of multimedia data. Those problemes for discrete (finite group) data where solved in the 90's. We show that all mentioned problems in real-valued data can be reduced to finding sparse solutions of systems of linear equations with rectangular matrices. It is known that in a general setting, the problem has non-polynomial computational complexity. The goal of this talk is to discuss existing sub-optimal methods and possible approaches in finding optimal methods. More about speaker, check here.

  • Talk Date: March 2, 2007

    Variation Models and PDE Techniques in Wavelet Inpainting

    Haomin Zhou

    Abstract:
    In this talk, I will present our work (collaborated with Tony Chan (UCLA) and Jackie Shen (Minnesota)) on image inpainting in wavelet domain. The problem is closely related to the classical image inpainting, with the difference being that the inpainting regions are in the wavelet domain, that brings new challenges to the reconstructions, as there is no geometrically well defined inpainting region in the pixel domain, and the damage is inhomogeneous. We propose new variational models, especially total variation minimization in conjunction with wavelets for the wavelet inpainting problems. The models lead to PDE's, which are Euler-Lagrange equations of the variational formulations, in the wavelet domain and can be solved numerically. The proposed models can have effective and automatic control over geometric features of the inpainted images including sharp edges, even in the presence of substantial loss of wavelet coefficients, including in the low frequencies. More about speakeri, check here.

  • Talk Date: 20070323

    Trading a mean reverting asset: buy low and sell high

    Qing Zhang

    Abstract
    This paper is concerned with an optimal trading (buy and sell) rule. The underlying asset price is governed by a mean-reverting model. The objective is to buy and sell the asset so as to maximize the overall return. Slippage cost is imposed on each transaction. The associated HJB equations (variational inequalities) are used to characterize the value functions. It is shown that the solution to the original optimal stopping problem can be obtained by solving two algebraic equations which are much simpler to solve. Sufficient conditions are given in the form of a verification theorem. A numerical example is reported to demonstrate the results. More about speaker, check here.

  • Talk Date: March 30, 2007

    High-order operator splitting methods for unitary and parabolic evolution

    Andrew Sornborger

    Abstract
    In this talk, I will discuss operator splitting methods and present methods that we developed for unitary quantum simulations on quantum computers. I will also discuss their application to parabolic differential equations and show that a conjecture that these methods are unstable in the parabolic case is incorrect. More about speaker, check here.

  • Talk Date: 20070406

    Darcy's Law and porous medium flow

    Ronghua Pan

    Abstract
    The motion of isentropic flow through porous media can be decribed by compressible Euler equtions with frictional damping. Time asymptotically, the density is conjectured to satisfy the porous media equation and the momentum obeys the classical Darcy's law. Previous results are valid for small smooth flow away from vacuum. In this talk, we report a proof for all physical flows. More about speaker, check here.

  • Talk Date: 20070420

    Approximation of functional regression models with bivariate splines

    Serge Guillas

    Abstract
    We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, with no noise. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with or without a penalization term. Under the assumptions that the regressors in the sample are bounded and span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations illustrate the quality of the asymptotic properties in various situations. More about speaker, check here.

  • Talk Date: 20070427

    SOME ASYMPTOTICS OF SELF-INTERACTING STOCHASTIC CHAINS

    Dan Kannan

    Abstract
    The protein folding, dark matter, chemotaxis, and polymers in solution are a few of the examples of self-interacting systems. Probabilists modeled, independently, some nearest neighbor interactions on lattices using random walks in random environmens, (recently applying such random walks to some of the above systems). We present, in this lecture, a more general foundation of self-interacting stochastic chains (SISC) and consider one or more of the following symptotic properties of the SISC: 1. Stability, 2. Functional limit theorems, and 3. Large deviation principle. More about speaker, check here.