Course Descriptions - Numerical Analysis/Scientific Computation
550.433 Monte Carlo Methods: The Monte Carlo method has proven to be an indispensable tool in any area of application involving stochastic modeling. The purpose of this course is to expose students to important ideas that arise when we employ the Monte Carlo approach. In the process, several key topics at the interface between numerical analysis, computing, probabilistic modeling, and statistics are covered, including: uniform random number generation, non-uniform random number generation, techniques for variance reduction, importance sampling, design of simulation experiments, Markov chain methods, applications to system reliability, and applications to error estimation for statistical methods. Prerequisites: 550.430, computing experience.
550.486 Asymptotic Methods: Methods for obtaining approximate analytical solutions to ordinary differential equations and difference equations. Topics vary depending on the instructor, but the course is likely to cover local analysis, asymptotic approximation, expansion of integrals, Laplace's method, Watson's Lemma, perturbation theory, summation of series, multiple scale analysis. Prerequisites: Calculus I & II and an introductory course in differential equations (550.291 or 550.303)
550.491 Applied Analysis for Engineers and Scientists: This course will cover techniques and applications of differential and integral analysis that are important for advanced work in engineering and science, including partial differential equations and transform methods. Prerequisites: Calculus 1, 2, 3, and either 550.291 and 500.303, or 110.201 and 110.302.
550.664 Modeling, Simulation and Monte Carlo: Concepts and statistical techniques critical to constructing and analyzing effective simulations; emphasis on generic principles rather than specific applications. Topics include model building (bias-variance tradeoff, model selection, Fisher information), benefits and drawbacks of simulation modeling, random number generation, simulation-based optimization, discrete multiple comparisons using simulations, Markov chain Monte Carlo (MCMC), and input selection using optimal experimental design. Prerequisites: basic matrix algebra and a graduate course in probability and statistics. Familiarity with some programming language such as Matlab, C, C++, or FORTRAN.
550.681 Numerical Analysis: Mathematical formulation and analysis of numerical algorithms. Brief review of topics in elementary numerical analysis such as floating-point arithmetic, Gaussian elimination for linear equations, interpolation and approximation. Core topics to be covered: numerical linear algebra including eigenvalue and linear least-squares problems, iterative algorithms for nonlinear equations and least squares problems, and convergence theory of numerical methods. Other possible topics: sparse matrix computations, numerical solution of partial differential equations, finite element methods, and parallel algorithms. Prerequisites: multivariable calculus, linear algebra, computing experience. Corequisite: 110.405.