Senior undergraduate courses

• 4315: Graph Theory
• 4320: Intro to Stochastic Processes
• 4331: Real Analysis
• 4350: Differential Geometry
• 4364: Numerical Analysis
• 4377: Advanced Linear Algebra

Online course

• 5330 Abstract Algebra
• 5331 Linear Algebra with Applications
• 5350 Into to Differential Geometry
• 5386 Regression and Linear Models

Graduate course

• 6302: Modern Algebra
• 6320: Real Variables
• 6324: Differential Equations
• 6342: Topology/Geometry
• 6360: Applicable Analysis
• 6366: Optimization Theory
• 6370: Numerical Analysis
• 6377: Basic Tools For Applied Math
• 6382: Probability Statistics
• 6384: Discrete Time Model in Finance
• 6395: Wavelet Analysis
• 6386: Computational Statistics
• 6397: Numerical Soltion Stochastic Diff Eqtns
• 6397: Duality Mthd and Operator Spaces
• 6397: Time Series Analysis
• 6397: Gibbs fields and their applications to automated Image Analysis (Cancel)
• 6397: Mathematical Hemodynamics
• 7320: Funtional Analysis
• 7396: Matrix Iterative Matheods

The course will be essentially self-contained but some of the material to be discussed can be found in:

• K.ATKINSON and W. HAN, Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd Edition, Springer-Verlag, New York, NY, 2005
• R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, 1984.

Following Part I of the course where various functional spaces of practical importance have been introduced, we will focus on the solution of variational problems from Image Processing (L^1 fitting, in particular). The following topics will be systematically discussed:

1. Existence and uniqueness of solutions to the variational problem.
2. Finite Element approximation.
3. Iterative solution by a variety of algorithms, including over-relaxation ones.

The methodology to be discussed can be applied to a variety of variational problems

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on floating point arithmetic, error analysis, interpolation, numerical quadrature, systems of linear and nonlinear algebraic equations.

Note: This is the first semester of a two semester course.

Recommended Texts

• Mathematical Statistics with Applications, Sixth Edition, by Wackerly, Mendenhall III and Scheaffer, Duxbury Press
• A First Look at Rigorous Probability Theory by Jeffrey Rosenthal, 2000.
• An Introduction to Stochastic Modeling, 3rd Edition, by H Taylor and S Karlin, Academic Press.
• A First Course in Probability, Sixth Edition by Sheldon Ross, 2002, Prentice Hall
• An Introduction to Probability Theory and Its Applications, Vol 1, 3rd edition, 1968 by William Feller (any edition would be fine).
• Probability by Leo Breiman, 1968, Addison- Wesley.

The main emphasis will be on numerical methods for the stochastic differential equations and Markov chains. In the beginning an overview of the necessary background in probability and stochastic DE's will be given. Very few formal proofs will be covered, but a brief overview of Probability, Markov processes, Wiener process, Ito calculus, Fokker-Planck equation, Ornstein-Uhlenbeck process will be provided.

Next, stochastic Taylor expansions, weak and strong convergence will be discussed. Stochastic schemes of various order and schemes based on exact solution will be discussed in detail. Students will have to complete programming assignments implementing some of these methods.

In the second half of the course the main emphasis will be on applications for stochastic differential equations. Relevant statistical quantities, numerical computation of distributions, equilibrium and non-equilibrium simulations, time- and ensemble averaging will be discussed. Applications of various techniques, such as Markov chain Monte Carlo, Adiabatic Elimination, Hidden Markov Models, Multiscale Methods, etc. will be presented.

None required. Texbooks that will be used are:

• W. Strauss's: "Partial Differential Equations"
• R. Glowinski: "Numerical Methods for Fluids (Part 3)"
• Chorin and Marsden: "Mathematical Introduction to Fluid Mechanics"
• Y.C. Fung: "Circulation"
• Y.C. Fung: "Biomechanics: Mechanical properties of living tissues
• R. LeVeques: "Conservation Laws", Research Papers

Review of basic linear PDEs. Analysis of quasilinear PDEs with concentration to hyperbolic conservation laws.

Introduction to fundamentals of fluid mechanics(basic equations of motion: continuity, momentum, energy, vorticity). Incompressible/compressible flow examples (derivation of the incompressible, viscous Navier-Stokes equations).

A brief introduction to Sobolev spaces. Fluid-structure interaction arising in blood flow modeling(effective models). Energy estimates.

Special topics related to the study of blood flow through compliant blood vessels.

For more infomation, click here

Reference books :

• Pierre Bremaud, "Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues" , Springer TAM Vol 31
• Bernard Chalmond, "Modeling and inverse problems in image analysis" Springer Vol 155, 2003

Gibbs fields are probabilistic models designed to analyze the behaviour of large systems of "particles" or "processors" in interaction ; they have been introduced by physicists for "statistical mechanics" describing the magnetisation of large nets of small magnets, and have generated exciting applications to modelize very large sets of digital images, in order to solve "low-level" artificial vision tasks , such as textures modelizations and segmentation, etc.

The main probabilistic topics impacting this course involve the dynamics of homogeneous and non-homogeneous Markov chains on (huge) finite spaces, Bayesian inference, efficient estimation of parameters in stochastic models , multidimensional Gaussian processes, Monte-Carlo simulations. We will present image analysis applications and sketch simulated annealing techniques for minimization of functions of very large numbers of discrete variables