AMSC/MAPL 460 or AMSC/MAPL 466, or knowledge of basic numerical
analysis (linear equations, nonlinear equations, integration,
interpolation) with permission of instructor. Knowledge of C or
Fortran or Matlab.
Course Description:
Monte Carlo simulation, numerical linear algebra, nonlinear systems
and continuation method, optimization, ordinary
differential equations. Fundamental techniques in scientific
computation with an introduction to the theory and software for
each topic.
basic numerical ideas
and programming practice: sources of error, conditioning and
stability, computer arithmetic, numerical resolution, order of
accuracy, accuracy check;
numerical differentiation, intergation, and
interpolation, error expansions and Richardson extrapolation.
Monte Carlo simulations
basic probability and statiscs: random variables, probability density
function (pdf), cumulative distribution function (cdf),
mean, variance, standard deviation, correlation, variance estimation,
confidence intervals,
central limit theorem. Markov process, etc;
basic Monte-Carlo simulation:
Pseudo random number generators, generation of nonuniformly
distributed numbers, Box-Muller methods for normals,
Brownian motion simulation and application to finance,
Monte-Carlo integration, rejection methods,
statistical analysis of simulation data and error bars;
variance reduction: stratified sampling, importance sampling, control variates, and antithetic variates
Numerical Linear Algebra
linear systems: estimation of errors, implementation using Matlab,
BLAS and block algorithms in LAPACK;
brief overview of least squares problem, data fitting, eigenvalue problems,
singular value decomposition
Optimization and Nonlinear systems
unconstrained optimization: Newton's method, line
searches (Golden section search, steepest descent)
Global optimization; safeguards in Newton's methods,
quasi-Newton methods, Davidson-Fletcher-Powell(DFP) method,
Broyden, Fletcher, Gordearb and Shannd(BFGS) method, Broyden family,
nonlinear conjugate gradient method (Fletcher-Beeves, Polak-Ribiere),
Memoryless quasi-Newton methods,
constrained optimization; quasi-Newton methods
System of nonlinear equations, Broyden method;
bifurcations: theorem and numerical treatment
stable and unstable branches
Ordinary Differential Equations
explicit and implicit methods, basic facts about stability and convergence;
GEAR-like packages. applications: combustion