Jonathan Goodman
Room 529, Warren Weaver Hall, NYU
goodman@cims.nyu.edu, 212-998-3326
Office Hours: Wednesday 4 to 6 pm, or by appointment
Course content
This is PhD level course on Monte Carlo methods. It is intended for mathematicians, comnputer scientists, scientists, statisticians, and others interested in learning about and using modern Monte Carlo methods in their research. The course covers basic sampling methods including mappings, rejection, and weighted direct sampling. Variance reduction methods include control variates, systematic sampling (Latin hypercube, Sobol), and importance sampling. Markov chain Monte Carlo (MCMC) methods, the basic Metropolis/heat bath/Gibbs sampler suite and modern improvements are addressed in detail. We discuss validation and error estimation for MCMC practice, including auto-correlation time and other convergence checks. Causes of slow convergence are identified, including collective modes, ill conditioning, and multi-modality. Advanced topics will depend on the interests of the students, but should include recent improvements in MCMC samplers, stochastic approximation and optimization, evaluation of evidence and partition function integrals, rare event sampling strategies, mathematical analysis of MCMC -- spectral gaps, burn-in time, etc. Applications in physical sciences, Bayesian statistics, and machine learning will be used.
Student project and evaluation
Students registered for the class are expected to participate in a semester long project. Students may work individually or in groups on a research project they are interested in. These may be developing a new MC method for an existing application, developing and experimenting with new samplers, trying a new application, or theoretical analysis. Most projects will lead to software that may be turned a package and shared. Students will give presentations on their project at the end of the semester. Successful projects may lead to a publication. There will also be regular assignments that involve computing and analysis.
Prerequisites
This is PhD level course. Students should have all these specific prerequisites:
- Probability: calculus based probability, multivariate densities, central limit theorem
- Linear Algebra: matrices, vector spaces, bases, eigenvalue/eigenvector decomposition for stability analysis of differential equations, SVD
- Programming in some language suitable for scientific computing and visualization (C/C++, FORTRAN, Python, Matlab, Java, ..
- Basics of scientific computing: numerical linear algebra and matrix factorizations, conditioning, FFT, multivariate Newton's method
Class communication
There is a page for the class on the NYU Classes system. This page will have a message board for student/student and student/instructor communication. Any registered student will have access. Contact the instructor for access to the site if you do not want to register for the course.