Differential equations and dynamical systems courses

This page collects some information about (ordinary) differential equations and dynamical systems courses offered at Caltech. This page was prepared in preparation for a faculty discussion on integrated ACM 106b, AM 125b and CDS 140a.

Overview of current course sequence

ACM 106b: Introductory Methods of Computational Mathematics

Catalog listing he sequence covers the introductory methods in both theory and implementation of numerical linear algebra, approximation theory, ordinary differential equations, and partial differential equations. The course covers methods such as direct and iterative solution of large linear systems; eigenvalue and vector computations; function minimization; nonlinear algebraic solvers; preconditioning; time-frequency transforms (Fourier, wavelet, etc.); root finding; data fitting; interpolation and approximation of functions; numerical quadrature; numerical integration of systems of ODEs (initial and boundary value problems); finite difference, element, and volume methods for PDEs; level set methods. Programming is a significant part of the course. Instructor: Yan. Dependent courses: ACM/CS 114 (Parallel Algorithms for Scientific Applications)? Partially overlapping courses AM 125b, CDS 140a

Topics (Winter 2010) Euler method Trapezoidal rule Multistep method Gaussian quadrature Runge-Kutta method Root finding problem Stiff equation (special class of ODE) Finite difference method Poisson equation (fast Poisson solver) Multigrid method Interpolation (approximation theorem)

ACM 216: Markov Chain, Discret Stochastic Processes and Applications

Catalog listing Stable laws, Markov chains, classification of states, ergodicity, von Neumann ergodic theorem, mixing rate, stationary/equilibrium distributions and convergence of Markov chains, Markov chain Monte Carlo and its applications to scientific computing, Metropolis Hastings algorithm, coupling from the past, martingale theory and discrete time martingales, rare events, law of large deviations, Chernoff bound Dependent courses ACM 217 Partially overlapping courses Ma/ACM 144 - Probability

Topics (Winter 2008) Markov Models. Transition matrices and Markov Chains. Kernels and Markov chains on arbitrary spaces. Finite Markov Chains. Markov Chains on countable state spaces. Simulations with Markov Chains. MCMC algortithms. Simulated annealing. The Propp-Wilson algorithm. Sandwiching. Rate of convergence

Advanced courses

There are several advanced courses that build on ACM 116/216 and are offered on a semi-regular basis:

• EE 164 - Stochastic and Adaptive Signal Processing
• ACM 217 - Stochastic Differential Equations and Applications
• ACM 217 - Stochastic Control Theory (taught once)
• ACM 256 - Large Deviation Theory and Concentration Inequalities (special topics course)