Michael Baron, (University of Texas at Dallas)

Friday, March 30, 2007

Optimal sequential detection of change points and Bayesian analysis of hidden Markov chains

John Neuberger (UNT)

February 23, 2007

A unified point of view on partial differential equations.

Abstract:  Will discuss how Sobolev gradients give both a computational and theoretical approach to the subject of partial differentials. This will be illustrated by applications to transonic flow, two-phase separation problems, elasticity and the Ginzburg=Landau functionals of  superconductivity.

Abstract:  In sequential change-point analysis, it is desirable to detect a change in distribution as soon as possible after it occurs while keeping the rate of false alarms to a minimum. Most change points occur unexpectedly, at random times, justifying the search for Bayesian procedures. Prior distributions of change point parameters may be constructed based on various surrounding factors (energy prices are likely to increase during a power plant maintenance), concurrent observation of another time series (high levels of air pollution, ozone, and pollen increase the chance of an influenza epidemic), or the observed process itself (after a long stable period of treatment, a patient is expected to show significant improvement). Often the whole prior distribution of a change point is not known, and only its discrete hazard rate function is available, i.e., the probability of a change point now, given that it has not occurred yet.  As often happens in sequential analysis, only in certain specific cases is it possible to construct Bayes sequential rules for change-point detection. Under suitable loss functions, Shiryaev (1978) solves the problem for the geometric prior distribution, and Ritov (1990) proves Bayesian property of the cusum scheme for certain specified prior.  The general form of a Bayes sequential rule has not been obtained. We derive the Bayes scheme for the situation when the hazard rate of a change point is determined by a homogeneous Markov process. In a more general setting, we propose asymptotically pointwise optimal procedures for any arbitrary prior. In all the cases, the risk functions are chosen to achieve an optimal balance between the mean delay and the mean time between false alarms. Sequential change-point detection schemes can also be used to analyze systems with multiple change points, hidden Markov chains, and situations when the first change may be followed by other changes or more complicated patterns.  Applications in epidemiology and energy finance will be discussed. 

Friday, January 26, 2007

Santiago Betelu

Recent developments in the study of electrically charged fluids in the talk "Electrically Charged Droplets".

The Applied Mathematics Seminar meets on Fridays at 01:00 pm, in room .  Everybody is invited to talk, regardless of your field of research(analysis, differential equations, teaching, probability, dynamical systems, differential geometry, algebra, etc.). The only constraints are: a) The talk must be related to an specific applied ("real world")problem, for example with applications to engineering, biology, material science, etc. b) It must be accessible to non-experts on the subject, including graduate students and people from other departments.''

Friday, January 26, 2007

Santiago Betelu

Recent developments in the study of electrically charged fluids in the talk "Electrically Charged Droplets".

The Applied Mathematics Seminar meets on Fridays at 01:00 pm, in room .  Everybody is invited to talk, regardless of your field of research(analysis, differential equations, teaching, probability, dynamical systems, differential geometry, algebra, etc.). The only constraints are: a) The talk must be related to an specific applied ("real world")problem, for example with applications to engineering, biology, material science, etc. b) It must be accessible to non-experts on the subject, including graduate students and people from other departments.''

Friday,  March 3, 2006

Prof. Tom Cundari from the Chemistry Department

Applications of Computers in Chemistry. A UNT Perspective

Abstract:  Computational chemistry is defined as the application of computer programs and algorithms to the modeling of chemical species and chemical processes. An overview will be given of the capabilities and facilties of University of North Texas' newly formed Center for Advanced Scientific Computing (CASCaM). The research strengths and interests of CASCAM faculty in the area of computational chemistry will be outlined. Areas for profitable interaction with applied mathematics will be sought.

Friday,  Feb.  17, 2006

Prof. Robert Renka

Title:  Curve Fitting with a Sobolev Gradient Method

Abstract:  Consider the problem of constructing a mathematical representation of a curve that satisfies constraints such as interpolation of specified points. This problem arises frequently in the context of both data fitting and Computer Aided Design. We treat the most general problem:  the curve may or may not be constrained to lie in a plane; the constraints may involve specified points, tangent vectors, normal vectors, and/or curvature vectors, periodicity, or nonlinear inequalities representing shape-preservation criteria. Rather than the usual piecewise parametric polynomial (B-spline) or rational (NURB) formulation, we represent the curve by a discrete sequence of vertices along with first, second, and third derivative vectors at each vertex, where derivatives are with respect to arc length. This provides third-order geometric continuity and maximizes flexibility with an arbitrarily large number of degrees of freedom. The free parameters are chosen to minimize a fairness measure defined as a weighted sum of curve length, total curvature, and variation of curvature. We thus obtain a very challenging constrained optimization problem for which standard methods are ineffective. A Sobolev gradient method, however, will be shown to be particularly effective.

Title:  Dimensionality Reducing by Alpha-Dense Curves: Application to Global Optimization, Multiple Integration and Mathematical Programming.

Gaspar Mora  Departamento de Analisis matematico de la Universidad de Alicante(Spain)

Title:  Filtering Out Noise to Fit a Prescribed Model

Abstract: Constructing a function by filtering noise from experimentally obtained data is a common problem in many areas of applied mathematics. In this work, we consider a filtering problem when the function is known to have certain theoretical properties which are masked by the noise. Convex quadratic programming is used to fit the closest fit to experimental data given the known theoretical constraints. The application of this technique to the modeling of aerogels with Gaussian random fields will be discussed. Quadratic programming has thus far provided superior fits in significantly less time than other standard fitting algorithms such as simulated annealing.

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Title:  SUPPORT VECTOR MACHINES DEMYSTIFIED

Robert R. Kallman

The subject of support vector machines (SVM's) is an approach to machine learning for decision-making and classification of objects which has attained an extremely popular, cult-like status in recent times. For example, searching the phrase "supportvector machine" with Google gives 290,000 hits, the phrase "support vector machines" gives 575,000 hits, the phrase "svm" gives 1,760,000 hits, and the phrase "kernel methods" gives 177,000 hits. However, unlike many cults there appears to be some definite merit to the subject with applications in diverse fields such as data mining. This has resulted in a flood of books, tutorials, and survey articles. All of these introductions to and explanations of SVM's are far more complicated and less general they need to be. For instance, no eigenvalue problems need to solved, no matrices need be inverted, no use of Mercer's Theorem is required, no Lagrangians need be  discussed, no general quadratic programming needs to be done, and no Kuhn-Tucker duality needs to be considered. This talk will describe in detail the theory behind a very elementary geometric approach to SVM's discovered this past summer. Only a rudimentary knowledge of real inner product spaces is required. Anyone who understands the theory can write a simple, efficient C-program to implement SVM's, as has been done. In addition, the subject of SVM's perhaps has some implications for pure mathematics in that it suggests a natural nonlinear generalization of the notion of hyperplane.

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Title: An Application of Optimization in Option Pricing

Jianguo Liu

Abstract: Options play a very important role in portfolio management. Hedging funds use options to make profits and mutual funds use options to manage risks. Option pricing is key to option trading. The well-known Black-Scholes equation in finance is the corner stone in option pricing. In order to determine a reasonable price for an option using the Black-Scholes equation, a parameter known as the volatility needs to be approximated in advance. We show some applications of optimization techniques in the determination of volatility.

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Title:  Computing the evolution of 3D charged droplets

Orestis Vantzos

We describe a numerical method for the computation of three dimensional flows of electrically charged fluids in droplets of microscopic scale. The motion is driven by capillarity and electrostatic repulsion. These flows are relevant to electrospraying and droplet breakup in storm clouds.  In this model, the fluid flow is described by the Stokes equations, and the electric potential by the Laplace equation on the exterior of the droplet. We solve the equations with a Boundary Elements Method on a triangulated grid that is dynamically adapted to refine in regions of high curvature and keep uniform triangulations. We show numerous results and discuss the usefulness and the limitations of the method.