Fundamentals of Uncertainty Quantification in Computational Science and Engineering
AM 238
Computing the statistical properties of nonlinear random systems is of fundamental importance in many areas of science and engineering. The primary objective of the course is to introduce students to state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of ordinary and partial differential equations. The course will focus mainly on the so-called forward UQ problem, in which uncertainties in input parameters such as initial conditions, boundary conditions, geometry or forcing terms are propagated through the equations of motion of the system into the solution. The course will also discuss cutting edge topics on data-driven modeling, deep learning with stochastic neural networks, and uncertainty propagation in high-dimensional systems via tensor methods.
Course Notes
| Title | Description | Date | Author |
|---|---|---|---|
| PDF equations for dynamical systems and PDEs | Course notes on PDF (probability density function) equations for analyzing the statistical properties of dynamical systems and PDEs evolving from random initial conditions. The topics include: Liouville equations for dynamical systems, systems with random parameters and colored noise, BBGKY PDF hierarchies, reduced-order PDF equations using conditional expectations, data-driven closure methods, Fokker-Planck equations, Hopf functional equations for PDEs, and Lundgren-Monin-Novikov hierarchies for studying turbulence statistics. | Nov 8, 2024 | |
| Polynomial chaos | Course notes on polynomial chaos theory and its applications to uncertainty quantification. Topics include Wiener-Hermite expansions for Brownian motion functionals, generalized polynomial chaos (gPC) expansions for systems with random variables, multi-element gPC methods, stochastic Galerkin projection techniques for solving random ODEs and PDEs (including heat equations, Burgers equations, and thermal convection), random eigenvalue problems, orthogonal polynomial theory, and convergence analysis of random variable sequences. | Nov 20, 2024 | |
| Probability spaces and random variables | Course notes on probability theory and random variables covering: probability spaces (sample space, event space, probability measure), σ-algebras, random variables and their mappings, cumulative distribution functions (CDFs), probability density functions (PDFs), functions of random variables with analytical transformations, applications to dynamical systems including the Liouville equation, sampling methods for arbitrary PDFs, and statistical moments/cumulants including expectation, variance, moment generating functions, and characteristic functions. | Oct 4, 2024 | |
| Random processes and random fields | Course notes on random processes and random fields, including stochastic process theory, Gaussian processes and their sampling methods, discrete Markov processes, particle filtering and Bayesian state estimation, Karhunen-Loève expansions, Wiener processes, and stochastic differential equations with their corresponding Fokker-Planck equations. The notes provide both theoretical foundations and practical computational methods for analyzing and simulating various types of random phenomena in continuous and discrete time. | Oct 22, 2024 | |
| Random vectors | Course notes on conditional probability distributions, Bayes’ theorem, and advanced sampling methods for high-dimensional probability density functions. The main topics include conditional expectations and correlations, Markov Chain Monte Carlo (MCMC) methods with emphasis on Gibbs sampling, and copulas for modeling dependence between random variables. | Oct 11, 2024 | |
| Sampling Methods | Course notes on sparse grids methodology for high-dimensional interpolation and integration. The topics include Chebyshev-Gauss-Lobatto grids, Smolyak interpolation algorithms, sparse grid construction and convergence analysis, integration on sparse grids with cubature formulas, Chebyshev polynomial theory, Lagrangian interpolation at Gauss points, and Lebesgue constants for error estimation. | Dec 4, 2024 |
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Course Work
The following are my solutions to the homework assignments presented to me in this class.
| Date | Title | Author |
|---|---|---|
| Dec 9, 2024 | Final Project | |
| Oct 2, 2024 | Homework 1 | |
| Oct 14, 2024 | Homework 2 | |
| Oct 23, 2024 | Homework 3 | |
| Nov 7, 2024 | Homework 4 | |
| Nov 14, 2024 | Homework 5 |
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