IAM768 - Special Topics: Methods and Applications of Uncertainty Quantification
Instructor(s): Consent of IAM
Prerequisites: Consent of Instructor(s)
Course Catalogue DescriptionProbability, 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.
Course ObjectivesStudents are expected to gain, besides theoretical concepts, programming skills that are related to Uncertainty Quantification and related applications.
Course Learning OutcomesBy 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.
Tentative (Weekly) OutlineProbability, 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.
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
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