Mohammad Motamed

MATH/STAT 579, Uncertainty Quantification

General description

The behavior of many physical systems can be mathematically modeled by deterministic ordinary or partial differential equations (ODEs or PDEs). These models however differ from reality for two main reasons: (1) an intrinsic variability of the physical system (aleatoric uncertainty); and/or (2) our inability to accurately characterize all parameters of the mathematical model (epistemic uncertainty). Uncertainty is therefore a fundamental feature of physical systems and needs to be taken into account when building a mathematical model. Examples appear in the description of flows in porous media, behavior of living tissues, combustion problems, deformation of composite materials, earthquake motions, etc. In order to understand the uncertainties inherent in the model, and often represented in a probabilistic setting, we need a process called uncertainty quantification (UQ).

In this course we consider different types of ODEs/PDEs with stochastic input parameters (coefficients, forcing terms, initial/boundary conditions, etc.) and study various numerical techniques for solving both forward and inverse stochastic problems.

Office hours

1) TBA SMLC 216 (Motamed)     2) TBA SMLC 314 (Huerta)


Lecture notes

  • Introduction: Lecture 1

  • Basics of Probability Theory: Lecture 2, Lecture 3

  • Karhunen-Loeve Expansion: Lecture 4

  • Forward UQ (Numerical Mathods for Stochastic ODEs/PDEs):
    • Intro and Abstract Framework: Lecture 5
    • Monte Carlo Smapling Mathods:
      • Classical Monte Carlo (MC): Lecture 6
      • Multi-Level Monte Carlo (MLMC): Lecture 7
      • Multi-Order Monte Carlo (MOMC): Lecture 8
    • Polynomial Approximation Methods:
      • Univariate and Multivariate Polynomials: Lecture 9
      • Stochastic Galerkin: Lecture 10
      • Stochastic Collocation: Lecture 11, Lecture 12, Lecture 13
      • Time-dependent PDE Problems: Lecture 14

  • Bayesian probability framework: Lecture 15, Lecture 16, Lecture 17

  • Maximum Likelihood: Lecture 18

  • Markov Chain Monte Carlo: Lecture 19, Lecture 20, Lecture 21, Lecture 22, Lecture 23, Lecture 24,

  • Gaussian Processes: Lecture 25, Lecture 26

  • Application to Climate Modeling: Lecture 27


Assignment 1 (Probability and Karhunen-Loeve)

Assignment 2 (Monte Carlo Sampling)

Assignment 3 (Polynomial Approximation)

Assignment 4 (Bayesian Inference)

Assignment 5 (MCMC)

Assignment 6 (Advanced methods)

Final Project
Last updated: August 2016