Probability, Random Processes, and Statistics; Markov Chains; Sampling and Monte Carlo Methods; Parameter Estimation; Uncertainty Propagation in Models; Stochastic Spectral Methods; Surrogate Models and Advanced Topics.

For further information see the academic catalog: IAM768

Course Objectives: Students are expected to gain, besides theoretical concepts, programming skills that are related to Uncertainty Quantification and related applications.

Course Learning Outcomes: By the end of this course, students should be equipped with fundamental methods of Uncertainty Quantification, and related concepts from Scientific Computing, Finance and Statistics, and Physics and Engineering.

Instructional Methods: Classical Lectures and Presentation of Students.

Course Website / Course Management System:

Weekly Outline / Tentative Course Schedule: 

  • Introduction and Preliminaries
    • Motivating Applications and Prototypical Models
    • Probability, Random Processes, and Statistics; Markov Chains
  • Sampling and Monte Carlo Methods
    • Computing Expectations/Integrals, Moments; Moment Approximations using Limit Theorems
    • Monte Carlo Methods, variance reduction techniques; importance sampling
  • Parameter Estimation
    • Frequentist Techniques: Linear Regression, Nonlinear Parameter Estimation, Optimisation and Algorithms (related content from least squares, regularization, etc.)
    • Bayesian Techniques: Markov Chain Monte Carlo, Metropolis-Hasting Algorithms, and Sequential Monte Carlo and Particle Filter; Delayed Rejection Adaptive Metropolis (DRAM), DiffeRential Evolution Adaptive Metropolis (DREAM)
  • Stochastic Spectral Methods
    • Orthogonal Polynomials, Piecewise Polynomial Approximation, Interpolation, Projection, (Gaussian) Quadrature Rules; Finite Elements (and, possibly, Finite Differences), Galerkin (Finite Element) Methods, (Polynomial) Spectral Methods
    • Spectral Expansion and Stochastic Spectral Methods: Karhunen-Loève Expansion, (generalised) Polynomial Chaos Expansion (gPC); Stochastic Galerkin Methods, Collocation, and Discrete Projection
  • Surrogate Models and Advanced Topics

Required Textbook/s & Readings:

  • Ralph C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2014
  • O. P. Le Maître, O. M. Knio, Spectral Methods for Uncertainty Quantification: with applications to Computational Fluid Dynamics, Springer, 2010
  • Mats G. Larson, Fredrik Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013
  • Steven M. Kay, Intuitive Probability and Random Processes using MATLAB, Springer, 2006
  • Paul Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003