Welcome to MATH 565 (Fall 2025)

• Office: RE 208
• Office hours: By appointment on Mondays 3:00 – 5:00 PM in RE 208 and Thursdays 9 – 11 AM on Microsoft Teams, email me if you need to make an appointment for another time
• Phone: 312-567-8983
• Email: hickernell@illinoistech.edu
• Website
• LinkedIn

• Google Scholar

• Brief bio: Fred J. Hickernell is professor of applied mathematics. His research focuses on increasing the efficiency of computer simulations and determining justifiable stopping criteria for simulation. A major area of interest is Monte Carlo methods.

  Hickernell’s research has been funded by the National Science Foundation and the Department of Energy. He is a Fellow of the Institute of Mathematical Statistics. In 2016, he received the Joseph F. Traub Prize for Achievement in Information-Based Complexity. He has served on the editorial boards of the Journal of Complexity, Mathematics of Computation, and the SIAM Journal on Numerical Analysis.

  Hickernell received his Ph.D. in mathematics from MIT and his B.A. in mathematics and physics from Pomona College. He came to Illinois Tech in 2005 as department chair and has also served as vice provost for research. Before coming to Illinois Tech, Hickernell was a professor in mathematics at Hong Kong Baptist University and assistant professor of mathematics at the University of Southern California.

  Hickernell speaks Cantonese and enjoys Chinese food. He is married with adult children. His most important identity is a disciple of Jesus.

• Office hours: Tuesdays and Thursdays, 9:30 – 11 AM in RE 119
• Email: rprasad1@hawk.illinoistech.edu

• VS Code
• Jupyter
• Github
• MATLAB
• LaTeX
• Overleaf
• qmcpy
• Course Repository

• Calculus
• Matrix algebra
• A calculus-based probability course, such as MATH 474 or MATH 475; you should understand
  • Discrete and continuous random variables
  • Probability mass and density functions, cumulative distribution functions
  • Mean, median, standard deviation, quantile, covariance, (in)dependence
  • Population versus sample quantities
  • Central Limit Theorem
• Facility in numerical programming, meaning
  • Programming in Python, or some other language such as MATLAB, or R
  • Using an integrated development environment (IDE), such as VS Code
  • You are highly encouraged to become familiar with GitHub
• Facility with LaTeX or some other technical document preparation system; Markdown is helpful

By the end of this course, students will be able to:

• Understand the basics of Monte Carlo and Quasi-Monte Carlo Methods.
• Understand the basics of Markov chain Monte Carlo (MCMC).
• Understand how these methods are used for computations.
• Assess the performance of Monte Carlo methods and improve their effectiveness.
• Understand basic implementation issues in performing Monte Carlo calculations.

How to Study for This Course

• What is a Monte Carlo method?
• Probability review
• Point and interval estimators of means, variances, probabililty distribution functions, and quantile functions
• Conditional Monte Carlo

• Pseudo-random numbers
• Random vectors with different distributions
• Low discrepancy sampling
• Accpeptance-rejection sampling

• Markov chains
• Metropolis-Hastings
• Kernel-based discrepancies and maximum mean discrepancy
• Discrepancy as a measure of cubature error
• Sample quality
• Gibbs sampler
• Convergence diagnostics
• Error estimation

• Control variates
• Importance sampling
• Antithetic variates
• Stratified sampling and Latin hypercube
• Quasi-Monte Carlo sampling

• (TBA)