MTH 654 - Sec 001
Numerical Analysis
Uncertainty Quantification

MWF 9:00-9:50AM
BEXL 328
Fall 2017


Professor:

Dr. Nathan Louis Gibson  

Office:

Kidd 056

Office Hours:

MWF 10-10:50AM

Course Website: 

http://math.oregonstate.edu/~gibsonn/Teaching/MTH654-001F17

 

Course Description

Background:

Accurate estimates for the propagation of uncertainty through complex systems is necessary for predictive simulation, robust design, and failure analysis. This course is primarily concerned with the numerical solution of differential equations which include uncertainty, either in system parameters, source terms, or in initial/boundary conditions.

Content:

In this course we will develop basic mathematical foundations and algorithmic aspects of stochastic computations and uncertainty quantification theory. The necessary background in polynomial approximation, numerical integration, and probability theory will be developed. While the emphasis will be on random differential equations, stochastic differential equations will be discussed. Methods covered will include Karhunen-Loeve expansion, generalized Polynomial Chaos, Stochastic Collocation, Spectral Stochastic Finite Element Method, Euler-Maruyama method for SDEs, among others. Topics will include convergence, stability, error estimates and implementation issues. As time allows, we will discuss challenges arising from high-dimensional problems, inverse problems/robust optimization and data assimilation.

Students:

The course will provide a working understanding of several numerical solution methods and MATLAB sample solutions to examples of applications. The course is intended for graduate students of mathematics, computer science, biology, science, and engineering. The content of the course is largely self-contained. For general pre-requisites, students should have some familiarity with probability, linear algebra, ordinary and partial differential equations, and introductory numerical analysis. Please contact the instructor with questions about background. Course requirements will be three homework assignments, which will consist of a mixture of analytical work and numerical computations with MATLAB, and a term project on a topic chosen by the student.

Text:

There is no required text for the course, but the following are recommended (in order of importance):
  1. Dongbin Xiu. Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, 2010.
  2. Ralph C Smith. Uncertainty Quantification: Theory, Implementation, and Applications, volume 12. SIAM, 2013.
  3. Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol 43, No 3, pp 525-546, 2001.
  4. T.J. Sullivan. Introduction to Uncertainty Quantification, Springer, 2015.
  5. Kloeden and Platen. Numerical Solution of Stochastic Differential Equations, Springer, 1999.
  6. Roger G Ghanem and Pol D Spanos. Stochastic Finite Elements: A Spectral Approach. Courier Dover Publications, 2003.
  7. Olivier P Le Maitre and Omar M Knio. Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer, 2010.

Grades

Grade Distribution

Homework 50%
Final Project50%
Total 100%

Matlab

Matlab is required for this course. Matlab is preferred due to the integration of computation and visualization.

Oregon State University has subscribed to a Total Academic Headcount (TAH) Site License for MATLAB. This new licensing includes many, but not all MATLAB toolboxes. OSU faculty, staff and students can install on up to 4 personally-owned devices or computers. For more information visit Information Services -- MATLAB or matlab.mathworks.com.

The following are online resources for learning Matlab:

Homework

Homework is required for this course. There will be a few short assignments, and they will be posted on the website. Problems will reinforce theoretical and computational concepts from lecture. Students are encouraged to work together, but must turn in individual papers.