Computational Science and Applied Math
Graduate Courses for Spring 2001

Math 579 - Singular Perturbations
Instructor:   C. Tier
Offered:   Spring Semester, 2001
Timetable:   08384 LECD 0100-0150 M W F 0305 TH
Office:   720 SEO ;    Phone:  312-996-2442 ;    Email:
Course Description:   (Bulletin) Algebraic and transcendental equations, regular perturbation expansions of differential equations, matched asymptotic expansions, boundary layer theory, Poincare-Lindstedt, multiple scales, bifurcation theory, homogenization.
Detailed Description:   The course deals with perturbation methods (regular and singular) which are used to systematically construct approximations to problems (e.g. ODE and PDE) that are otherwise intractable. The approximations are analytic and often provide important insight into the behavior of the solution. This is in contrast to numerical solutions which can provide extremely accurate results yet not provide sufficient information about the solution. The purpose of the course is to survey important perturbation methods. These methods will be contrasted with numerical methods and it will be illustrated how the perturbation and numerical methods often complement one another.
Course Web Page:
Text:   M. H. Holmes,  Introduction to Perturbation Methods,  Springer-Verlag, 1995.
Reference:   E. Bender and D. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Wiley.
Tentative Topics:
  • Introduction to asymptotic approximation
  • Matched asymptotic expansions; boundary layer expansions
  • Method of multiple scales - slowly varying coefficients
  • WKB method
  • Bifurcation and stability
Prerequisites:   Background in differential equations (ODE and PDE) or consent of instructor.

Math 590 Advanced Topics in Applied Mathematics: Computational Finance
Instructor:   C. Tier
Office:   720 SEO ;    Phone:  312-996-2442 ;    Email:
Offered:   Spring Semester, 2001
Timetable:   62915 LECD 0300-0415 M W 0208 TH
Financial derivatives are a multi-trillion dollar component of today's financial markets and often play a key role in complicated financial transactions. Because of their importance, the study of financial derivatives has grown dramatically in the financial industry. The course will present current topics in computational finance including pricing of derivative instruments such as options, interest rates and other contracts. The emphasis will be on the computation of fair market prices and other quantities of interest. This will involve the solution of partial differential equations that arise from stochastic models. Analytic methods will be used if possible. Numerical methods applicable to intractable models will be stressed. Computation tools such as Maple will be used along with programming in higher level languagues.
Comments:   This course is part of the Computational Finance track in the MISI program in MSCS.
Prequisites/background:   Students should be familiar with basic probability, differential equations and elementary numerical methods.
  1. Introduction to derivatives - European and American stock options, combinations to options, replication of contracts, valuations and profit and loss curves, arbitrage and the principle of non-arbitrage pricing.
  2. Stochastic models of assets: theories of Bachelier, Black-Scholes, Merton.
  3. Brief review of partial differential equations; backward and forward diffusion equations, analytic solution of Black-Scholes model; free boundary value problem for pricing American options, similarity solutions
  4. Numerical solution of partial differential equations arising in pricing models; finite difference methods - explicit and implicit, Crank-NIcholson, methods for American options.
  5. Pricing exotic options - barrier, Asian and Lookback options
  6. Interest rate models - yield curves, bond pricing models, parameter fitting.
  7. Quasi-Monte Carlo numerical methods, if time permits.

Math 590 Advanced Topics in Applied Mathematics: Mathematical Finance
Instructor: S. Yau
Offered:   Spring Semester, 2001
Timetable:   62907 LECD 1100-1150 M W F 0303 AH

MCS 504 - Mathematics and Information Science for Industry Workshop.
Instructor:   R. Grossman
Text:   None; selected articles will be used.
Offered:   Spring Semester, 2001
Timetable:   63931 LECD 0200-0500 F 0700 SEO
Course Description:   This course is centered around one or more "industrial" problems. The goal of the course is to provide an opportunity for students to use mathematics and information sciences to work on problems arising from industrial applications. The course will cover: mathematical modeling, problem formulation, problem analysis, problem solution, developing software to implement the solution, validating the software, analyzing the results, documenting the problem and its solution, techniques for effectively working in groups, software engineering, and effectively communicating technical material.
Comments:   The course may be repeated for credit.
Prerequisites:   Prior course work in data structures and algorithms and C/C++ programming

MCS 507 - Mathematical, Statistical & Scientific Software
Instructor:   F. Hanson
Text:   (Background Reference to Prof. Hanson's Lectures): William H. Press et al., Numerical Recipes in C: The Art of Scientific Computing, Cambridge U. Press, ISBN: 0521431085 ( price: $57.95; also available for F77 and/or F90; Disk of programs available in Fortran and nonstandard fortran translated C). (NRC)
Offered:   Spring Semester, 2001
Timetable:   63945 LECD 0430-0545 M W 0202 LH
Course Description:   (Bulletin) The design, analysis and use and of mathematical, statistical, and scientific software.
Comments:   This is a new course that is a core course in the Master of Science Degree program in Mathematics and Information Sciences for Industry, which became official Fall 2K.
Prerequisites:   Grade of B or better in MCS 471, an equivalent course, or consent of instructor.
The MCS 507 , web page has further information for this course.

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