Monte Carlo Methods

Course description

A graduate class on Monte Carlo methods. It is a combination of basic material and more advanced topics. The basic material includes sampling methods and error bars, variance reduction method (importance sampling, control variates), rejection and Markov chain Monte Carlo (MCMC). Advanced topics subject to change, but probably include

• Thermodynamic integration with physical applications and also application to Bayesian model selection. The ideas, in different forms, arose independently in computational physics (for computing free energy and partition functions), in computer science (for computing volumes of convex bodies), and in engineering (for estimating failure probabilities of reliable systems).
• Rare event simulation. Some of the methods involve a combination of importance sampling and theoretical rare event analysis. Others are more closely related to thermodnamic integration.
• More modern Markov Chain Monte Carlo methods, including Langevin and Hamiltonian strategies, affine invariant and ensemble samplers, and samplers for distributions with constraints.
• Theoretical analysis of MCMC efficiency, including the small set Lyapunov function method, the Lovasz Simonovic version of Cheeger's inequality and conductance, and Poincare inequalities.

Depending on student interest and time, other possible topics include

• Eigenvalue estimates (electronic structure) from quantum chemistry.
• Stochastic optimization methods such as Robbins Monro (stochastic approximation), and stochastic gradient decent and related methods,
• Methods for simulation of stochastic differential equations.

There will be much theoretical material, but everything will be illustrated in practical applications from diverse fields.

Prerequisites:

The ability to write computational programs, preferably in Python. A basic course in probability that includes multi-variate densities and the multi-variate central limit theorem. A good background in linear algrbra and multi-variate calculus. Preferably a course on scientific computing or numerical analyis.

Assignments and work flow

This COVID impacted class will be mostly remote with some in person class time depending on student interest and how things go at NYU this September. The following learning and workflow schedule follows math department recommendations for graduate courses this fall.

• There will be regular weekly classes on zoom (and possibly also in person after a few weeks) from 5:10 to 7 pm each Wednesday.
• Before this class you should read the posted lecture notes for the week and watch the posted videos. These will contain all the course material for the week. This system is not in place for week 1, but will start for week 2.
• After the notes and videos, but before chall class, you should do the weekly "I read it" online quiz. This will be on the NYU Classes site for the course. This system is not in place for week 1, but will start for week 2.

Group final project

Students will work in small gruoups (up to 4 members, or permission of the instructor) to do a research project that explores some aspect of Montey Carlo. Each group will prepare a "written" (PDF file) writeup of their project and make a presentation to the class. Project organization (group formation, topic selection, etc.) will begin after week 4.

Web sites:

There are two web sites for the class, a public site (this one) and an NYU Classes site for the class. Educational materials will be posted on the public site. The weekly quizzes and homework upload mechanism is on the NYU Classes site. The Classes site also will have a communication forum and access to your entries in the gradebook.

Assignments, exams, grading:

The final course grade will be determined by a weighted sum of scores for quizes (5%), assignments (65%), and the final project (30%). I try to use the gradea A, A-, B+ and B, with lower grades only for people who "earn" them by failing to do much of the assigned work. Students who make a good faith effort should not expect a grade below B. Please contact me immediately if the material or the assignments are unmanageable, particularly if you are weak in some of the prerequisites.

Communication:

Please use the Forum page of the NYU Classes site for this course for all content related communication, including questions about assignments, lectures, or notes. Feel free to contact the instructor directly about other issues such as appointments, missed classes, late assignments, grading issues, etc. The instructor and TA will check the message board frequently. Look there for important course announcements, in particular corrections to assignments.

Academic integrity

Please review the academic integrity policies of the math department and the Graduate School of Arts and Sciences. The policies for this course are

• Students are encouraged to cooperate with each other in learning the material and figuring out how to do the assignments.
• Students must create all code and assignment writeups individually. Code sharing is not allowed. Downloading code or recieving outside coding help is not allowed. Students are not allowed to copy writeups from other students or have help in writeups from web resources or outside parties. Anyone who gives code or writeups, or recievel them, is in violation of the integrity policy. Students who violate the policy may recieve punishments such as grade reduction or being reported to the department or graduate school.
• I urge students to report violations they suspect or become aware of.
• If the workload is so heavy that it is imnpossible to do in a reasonable amount of time, please report that to me as soon as possible.