MATH 114/214: Introduction to Computational Stochastics
Spring quarter, 2021

Course Description This is an introductory course on computational stochastic processes. Topics include sampling random variables, variance reduction, Markov chain Monte Carlo simulations, analysis of simulation data, numerical methods for stochastic differential equations, and applications in physics, chemistry, and biology.

The course is co-listed as Math 114 (for undergraduate students) and Math 214 (for graduate students). A course project is part of Math 214.
       

Prerequisites The prerequisite for Math 114 is Math 180A. Students who have not completed the prerequisites may enroll with consent of instructor.

In addition to the designated prerequisites, students are strongly recommended to have completed a computer programming course before enrolling in Math 114. Examples of such a course include: CSE 5A, CSE 8A, CSE 11, or ECE 15.
       

Lecture Notes All lecture notes will be posted in Canvas.
Textbooks We will use the following two books. UCSD has the full access to these eBooks.
  • R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method, 3rd ed., John Wiley & Sons, 2017.
  • Emmanuel Gobet, Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear, Chapman and Hall/CRC, 2020.

References Here is a list of reference books; UCSD has the electronic access to many of these books.
  • G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 3rd ed., 2001.
  • G. F. Lawler, Introduction to Stochastic Processes, 2nd ed., Chapman & Hall/CRC, 2006.
  • M. A. Pinsky and S. Karlin, An Introduction to Stochastic Modeling, 4th ed., Acdemic Press, 2010. (UCSD has access to the ebook.)
  • S. Asmussen and P. W. Glynn, Stochastic Simulation, Springer, 2007.
  • D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods, John Wiley & Sons, 2011.
  • J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2004.
  • C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, 2004.
  • N. Madras, Lectures on Monte Carlo Methods, Amer. Math. Soc., 2002.
  • C. Graham and D. Talay, Stochastic Simulation and Monte Carlo Methods, Springer, 2013.
  • P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992.
Topics to Be Covered
  • Introduction
    • Four examples
    • Outline of the course
  • Sampling random variables
    • Concept, examples, and data processing
    • The inversion method and the transformation method
    • The acceptance-rejection method
    • Sampling Bernoulli, bionomial, geometrical, and Poisson variables
    • Sampling random points on a geometric object
  • Variance reduction
    • Controlling variables
    • Common-variable and antithesic-variable methods
    • Importance sampling
    • Stratified sampleing
  • Markov chain Monte Carlo
    • Concept, examples, and basic properties of Markov chains
    • The Metropolis-Hastings algorithm
    • The simulated annealing algorithm
    • The Gibbs sampler
  • Numerical stochastic differential equations (SODE)
    • A brief introduction to Brownian motion
    • Simulation Brownian motion and its variants
    • A brief introduction to Ito calculus and SODE
    • Euler-Maruyama and Milstein's methods
Lectures 11:00 am - 11:50 am, Mondays, Wednesdays, and Fridays, remote. (Lectures will be given via Zoom. The Zoom lectures will be recorded in order to accommodate students in different time zones.)
Instructor Professor Bo Li.
Email: bli@math.ucsd.edu
Office hours: 4:30 pm - 5:30 pm, Mondays and Thursdays, remote.
Discussion Sessions Session A02, 4:00 pm - 4:50 pm, Tuesdays, remote.
Session A01, 5:00 pm - 5:50 pm, Tuesdays, remote.
Teaching Assistant Mr. Zirui (Ray) Zhang
Email: zzirui@ucsd.edu
Office hours: 10:00 am - 12:00 noon, Tuesdays, and 4:00 pm - 6:00 pm, Wednesdays. Remote.
Canvas and GradeScope Canvas will be used to manage our course (e.g., scheduling Zoom lectures and office hours, assigning homework, etc.). GradeScope will be used to collect and grade homework and exams, and manage the grding.
Homework Homework is assigned collected, and partially graded regularly. A large portion of homework is computationally oriented. Students can use Matlab or other computer software and languages.

Please note:

  • No late homework will be accepted unless a written verification of a valid excuse (such as hospitalization, family emergency, religious observance, court appearance, etc.) is provided.
  • While discussions are allowed, students must complete their homework assignments independently. Copying homework (entirely or partially) is absolutely prohibited.
Here are the homework assignments and their due times:
Exams and Projects For Math 114: There will be one 50-minute midterm and one 3-hour final exam, both synchronous and conducted via Zoom. The final exam will be cumulative.
      Midterm Exam: 11:00 am - 11:50 am, Friday, April 30, 2021.
      Final Exam: 11:30 am - 2:29 pm, Friday, June 11, 2021.

For Math 214: There will be a 50-minute, synchronous, midterm exam and a final course project.
      Midterm Exam: 11:00 am - 11:50 am, Friday, April 28, 2021.
      The final course project is due 3:00 pm, Friday, June 11, 2021.

Please note:

  • Neither rescheduled nor make-up exams will be allowed unless a written verification of a valid excuse (such as hospitalization, family emergency, religious observance, court appearance, etc.) is provided.
  • For all the exams, students can use the textbook, lecture notes of the instructor and class notes of students own, and students own homework solutions, all of this course. No other material (e.g., anything else from the internet) should be used, and no discussions are allowed. Questions should be placed in the Zoom chat room.
  • No previous research project or course project from a different course will be accepted as a final course project.
Project Guidelines This is only for Math 214. Here is a set of guidelines for course projects (requirement, some suggested topics, etc.): Project Guidelines.
Grading Math 114: The final course grade will be determined based on the homework (40%), midterm exam (20%), and final exam (40%). The final course grade will be curved within Math 114.

Math 214: The final course grade will be determined based on the homework (40%), midterm exam (20%), and final course project (40%). The final course grade will be curved within Math 214.
       

Academic Integrity All students are expected to conduct themselves with academic integrity. Violations of academic integrity will be treated seriously. See
Last updated by Bo Li on May 15, 2021.