# Math 577 -- Monte Carlo Methods

## Spring 2022

 Instructor: Kevin Lin Time: TTh 1100-1215 Place: Math 514 402 MTL 124 (note room change AGAIN!) Office: Math 606 Office hours: see here Phone: +1-520-626-6628 E-mail: klin@math.arizona.edu Course web page (this page): http://www.math.arizona.edu/~klin/monte-carlo

### Announcements

```* <2022-01-12 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.

* <2022-01-12 Wed> Room change
Due to capcity issues, we will meet in Math 402 for the rest
of the semester.
```

### Syllabus

```* <2022-01-13 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 higher-dimensional 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
Examples will be drawn from physics, chemistry, Bayesian statistics, and other fields, depending in part on student interest. However, no background in these areas is assumed. The second part covers more specialized topics. Potential topics may include (but are not limited to)
• Sequential Monte Carlo, filtering
• Rare event simulation
• Gillespie & related algorithms
• Exact sampling
A detailed week by week plan will be posted on this page, and updated as we go along.

### 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 pencil-and-paper analysis and computer experiments.

Interested students unsure about their background are encouraged to see the instructor prior to registering for the course.

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
There are no exams in this course.

### 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.