Math 577  Monte Carlo Methods
Spring 2019
Instructor:
Kevin
Lin
Time: WF 12001315
Office: Math 606
Course web page (this page):

Announcements
(Last revised on November 05, 2019.)
*fillcolumn:65536* * Student papers  Brian Bell, A sampling method for detecting adversarial examples  Sara Bredin, Identifying gerrymandering through monte carlo methods  Loren Champlin, Finding counterfactual explanations using a modified simulated annealing  Kevin Luna, Stochastic modeling of chemical kinetics  Jessica Pillow, MCMC and quantitative image reconstructions  Craig Thompson, Pixel clustering with a Gaussian mixture model: Two algorithms  Ruiyang Wu, Introduction to exact sampling  Shenghao Xia, MCMC on neuron spike sorting model  Zhuocheng Xiao, Importance sampling for integratefire equations I have permission to share some of the papers. Plealse contact me (or the author!) directly if you are interested. * old announcements
Course description
Monte Carlo methods are numerical algorithms that use random sampling to generate statistical estimates of (usually) deterministic quantities. They are often more efficient than their deterministic counterparts, especially for higherdimensional problems, and are widely used in scientific, engineering, and statistical computing.This introductory course is aimed at graduate students in mathematics, statistics, computer science, engineering, physical sciences, quantitative biology, or really any field where Monte Carlo methods are used. The goal is to equip students with knowledge of basic algorithms and relevant theory so they can design and implement Monte Carlo solutions to scientific problems and perform basic statistical analysis on the output. As such, the course tries to balance between discussion of practical algorithms and their mathematical analysis.
The first part of the course will cover
 Direct sampling methods
 Markov chain Monte Carlo
 Error analysis
 Variance reduction
 Importance sampling
 Sequential Monte Carlo, filtering
 Rare event simulation
 Gillespie & related algorithms
 Stochastic differential equations
 Exact sampling
Prerequisites
Students should know probability at the undergraduatelevel, e.g., Math 464 or equivalent, as well as linear algebra (e.g., Math 410 or equivalent). Some of this material will be briefly reviewed, and additional mathematical tools (e.g., theory of Markov chains) will be covered as needed. Students are expected to carry out both pencilandpaper analysis and computer experiments, so facility in or willingness to learn a programming language (e.g., Matlab, Python, R, Julia, ...) is required.Interested students without this background are encouraged to see the instructor prior to registering for the course.
Grading
Grading will be based on a few problem sets (due every 23 weeks) and an individual term project. Students will consult the instructor to choose a suitable project topic; projects related to students' own research are especially encouraged. Project results will be presented to the class and summarized in a term paper, to be published on the course web pageThe grade breakdown is as follows:
 40% homework
 30% project paper
 20% project presentation
 10% participation
Syllabus
These are the topics I'm planning to cover. It(Last revised on March 12, 2019.)
* PART I: Fundamentals  Direct methods, e.g., transform and rejection methods  Markov chain Monte Carlo (MCMC) 1. Basic Markov chain theory 2. Some standard MCMC algorithms 3. Error analysis  Variance reduction, e.g., control variates, antithetic variates.  Importance sampling * PART II: Selected Topics  Rare event simulation  Sequential MC and filtering
References
There is no required text, and I do not plan to follow any one text closely. I do plan to assign reading from some of the books and lecture notes listed below, in addition to course notes and papers to be posted later in the term.All these are either downloadable for free from the UA Library (just follow the links below from a computer on the campus network, or use VPN), or available from the authors.
The following standard textbooks on Monte Carlo are intended as primary references. The mathematical level of the course is somewhere between [RK] and [Liu]; [KW] is a good source for some of the topics we'll discuss.
 [KW]
Monte
Carlo Methods by MH Kalos and PA Whitlock
 [Liu]
Monte
Carlo Strategies in Scientific Computing by JS Liu
 [RK]
Simulation
and the Monte Carlo Method, Third Edition by RY Rubinstein and DP
Kroese
 [AG] Stochastic
Simulation: Algorithms and Analysis by S Asmussen and PW Glynn
 [HH] Monte
Carlo Methods by JM Hammersley and DC Handscomb
 [O] Monte
Carlo theory, methods and examples by AB Owen
 [M]
Introduction to
Monte Carlo Methods by DJC MacKay
 [S] Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms by A Sokal, in Functional Integration: Basics and Applications, C DeWittMorette, P Cartier, and A Folacci, eds. NATO ASI Series (Series B: Physics) 361, Springer, 1997
For sequential Monte Carlo, particle filtering, etc., see
 [DFG] A Doucet, N Freitas, and N
Gordon, Sequential
Monte Carlo Methods in Practice, Springer (2001)
 [vLCR] PJ van Leeuwen, Y Cheng, and S Reich,
Nonlinear
Data Assimilation, Springer (2015)
 [Evensen] G
Evensen, Data
Assimilation: the Ensemble Kalman Filter, Springer (2009)
 [HPS] Introduction to Stochastic Processes by Hoel, Port, and Stone
This page was last updated on April 27, 2019.