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AMS 691, Topics in Applied Math 

Varying topics selected from the list below if sufficient interest is shown. Several topics may be taught concurrently in different sections: Advanced Operational Methods in Applied Mathematics, Approximate Methods in Boundary Value Problems in Applied Mathematics, Control Theory and Optimization Foundations of Passive Systems Theory, Game Theory, Mixed Boundary Value Problems in Elasticity, Partial Differential Equations, Quantitative Genetics, Stochastic Modeling, Topics in Quantitative Finance.


AMS 691.01 Topics in Applied Mathematics:
This is a three credit intermediate graduate course on the finite element methods, their applications in engineering and science, and their recent trends in research.  It targets graduate students in applied mathematicians, computer science, engineers, and physics.

This course consists of two parts.  The first part is more user-oriented, which will introduce the mathematical formulation of finite element methods for solving partial differential equations, pre- and post-processing, as well as their use cases in fluid dynamics, structural mechanics, and multiphysics problems.  The open-source software FEniCS and the commercial software ANSYS will be used as demonstrations.  The second part is more mathematical and research-oriented, which will cover the analysis of accuracy and stability, efficient implementations and solvers, limitations of traditional FEMs, and more recent development in the extensions of FEMs and their unifications with other numerical methods.

Computing projects will be involved.  Python and/or Octave will be used as the programming environments for this course.

Prerequisites:  Multivariable calculus and linear algebra (e.g. AMS 510), numerical analysis (e.g. AMS 527), prior programming experience in MATLAB, Python, or C/C++ (e.g. AMS 595/ 561); or pre-approval by the instructor.  A laptop (64-bit Windows, Mac, or Linux) is required.
AMS 691.01 Topics in Applied Mathematics (Interest Rate and Credit Modeling):
Introduction to most commonly used interest rate models: Heath-Jarrow-Morton, Brace-Gatareck-Musiela, etc. Cap, Floor, Euopean and Bermudan option pricing.  Credit modeling: Merton structural approach vs. Intensity approach.  Corporate bonds, CDS, securitized products (CDO, CLO, mortgages), Credit value adjustment (CVA, XVA).
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