Math 577  Monte Carlo Methods
Spring 2022
Instructor:
Kevin
Lin
Time: TTh 11001215
Office: Math 606
Course web page (this page):

Announcements
* <20220112 Wed> Zoom I plan to stream + record the first couple weeks of classes (and perhaps more), in case anyone needs to (or would just like to) attend remotely. Stay tuned for updates. * <20220112 Wed> Room change Due to capcity issues, we will meet in Math 402 for the rest of the semester.
Syllabus
* <20220113 Thu> Introduction
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 and applications
 Markov chain Monte Carlo
 Error analysis
 Variance reduction
 Importance sampling
 Sequential Monte Carlo, filtering
 Rare event simulation
 Gillespie & related algorithms
 Exact sampling
Prerequisites
Students should know probability at the advanced undergraduate level, e.g., MATH 464 or equivalent, as well as linear algebra (e.g., MATH 410 or equivalent). I plan to briefly review some of this, and cover any additional material (e.g., Markov chains) as needed. Students are expected to carry out both pencilandpaper analysis and computer experiments.Interested students unsure about their background are encouraged to see the instructor prior to registering for the course.
Grading
Grading will be based on a small number of problem sets 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 page after the end of semester. Because term papers will be published, you should be careful not to include new work that you plan to publish later.The grade breakdown is as follows:
 40% project paper
 30% project presentation
 20% homework
 10% participation
References
There is no required text, and I do not plan to follow any particular 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. Of these, [O] is perhaps closest in spirit and level to this course.
 [KW]
Monte
Carlo Methods by MH Kalos and PA Whitlock
 [K]
Course notes
on Monte
Carlo Methods by DP Kroese
 [Liu]
Monte
Carlo Strategies in Scientific Computing by JS Liu
 [O] Monte
Carlo theory, methods and examples by AB Owen
 [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
 [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)
 [D] Essentials of Stochastic Processes by Durrett
 [HPS] Introduction to Stochastic Processes by Hoel, Port, and Stone