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) undergraduate analysis (Math 35/63), (ii) linear algebra (Math 22/24), (iii) probability/statistics (Math 20/40/60 or 10/70) 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 (40%), (ii) Midterm (20%), (iii) Final exam (40%). Homework will include theory and computer simulation problems.
The following plan is subject to further changes.
|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|
|Week 5||Stochastic Processes|
|Day 1||Brownian Motion|
|Day 2||Stochastic Differential Equations|
|Day 3||Stationary Stochastic Process|
|Week 6||Polynomial Chaos|
|Day 1||Karhunen-Loeve Expansion|
|Day 2||Generalized Polynomial Chaos|
|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|