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: odtuclass.iam.metu.edu.tr

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