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 Markov chain Monte Carlo (MCMC). We discuss validation and error estimation methods, including auto-correlation time for MCMC. We will discuss variance reduction methods, such as control variates, systematic sampling, and importance sampling. 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.
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
Expectations and grading
There will be some assignments during the term and a larger final project. Students are encouraged to do the project in small groups. PhD students who have passed their departmental oral exams and are engaged in PhD research may be able to negotiate over the workload.
There is a page for the class on the NYU Blackboard 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.