MATH 126 Topics in Applied Mathematics: Data-driven Uncertainty Quantification

Course Description: Uncertainty quantification is central to the study of science and engineering that involves unknown parameters and random behaviors. This course introduces theories and methods in uncertainty quantification, in particular, data-driven methods, which find applications in data science, machine learning, and numerical weather prediction. As computational tools are essential in uncertainty quantification, the course also introduces standard computation libraries and involves coding in MATLAB/Python.

Prerequisites: (i) linear algebra (Math 22/24) or permission of the instructor.

Textbook: There is no textbook for this course. As supplemental materials, the following books are recommended but not required to purchase.

Grading Formula: (i) Homework (60%), (ii) Final exam (40%). Homework will include theory and computer simulation problems.


  • Campus licencse MATLAB can be downloaded here.
  • Scientific computing packages for Python can be downloaded here.
  • Syllabus

    The following plan is subject to further changes.

    Week 1 Introduction
    Day 1 Uncertainty and data-driven quantification
    Day 2 Review of probability
    Day 3 Information Theory
    Week 2 Statistical Inference
    Day 1 Parametric inference
    Day 2 Nonparametric inference
    Day 3 Bayesian inference
    Week 3 Random Sampling
    Day 1 Monte Carlo
    Day 2 Importance sampling (Matlab/Python Code)
    Day 3 Markov chain Monte Carlo (Matlab/Python Code)
    Week 4 Special Topics
    Day 1 Hilbert Space
    Day 2 Smoothing using Orthogonal Functions
    Day 3 Midterm
    Week 5 Stochastic Processes
    Day 1 Brownian Motion
    Day 2 Stochastic Differential Equations
    Day 3 Stationary Stochastic Process, Stock price data
    Week 6 Polynomial Chaos
    Day 1 Karhunen-Loeve Expansion
    Day 2 Generalized Polynomial Chaos
    Day 3 Review
    Week 7 Data Assimilation
    Day 1 Kalman Filter (Matlab code)
    Day 2 Approximate Gaussian Filters I
    Day 3 Approximate Gaussian Filters II
    Week 8 Advanced Data Assimilation
    Day 1 Ensemble Square Root Filter
    Day 2 Particle Filter (particle filter Matlab code for Lorenz 63)
    Day 3 Localization and Inflation
    Week 9 Challenges of High-dimensional Spaces
    Day 1 Sampling in High-dimensional Spaces
    Day 2 Data Assimilation in High-dimensional Spaces
    Day 3 Summary

    Course Policies

    Honor Principle

    Collaborations (giving and receiving assistance) during closed-book exams and quizzes are strictly prohibited. Any form of plagiarism is not allowed in the final project. If you have questions, please ask the instructor before doing and should always refer to Academic Honor Principle.

    Accessibility Policy

    Students with learning, physical, or psychiatric disabilities enrolled in this course that may need disability-related classroom accommodations are encouraged to make an office appointment to see your instructor before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested. At such a meeting please provide your instructor with a copy of a disability registration form, which lists the accommodations recommended for the student by Student Accessibility Services within the Academic Skills Center. The person you might want to contact at the Academic Skills center is Ward Newmeyer, Director of Student Accessibility Services 205 Collis Center - (603) 646-9900.

    Student Religious Observances

    Some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with your instructor before the end of the second week of the term to discuss appropriate accommodations.

    Late Policy

    By "deadline" we really mean it. In exceptional circumstances, students with disabilities should inform the instructor of their accommodation requests well in advance, so that the instructor will have sufficient time to work with Student Accessibility Services to provide appropriate accommodations.