Stochastic numerical methods in uncertainy quantification

Garant předmětu:

Ivana Pultarová

Lecturer:

Ivana Pultarová

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Students are introduced to basic solution methods for problems dependent on random input variables and for estimating of models and their parameters from measured data. The subject focuses on the computational properties of these methods, related numerical methods, their convergence conditions and efficiency. The particular topics are numerical solution of determinic partial differential equations, finite element method, finite difference method (only a sketch of them both); basic methods of computational probability; partial differenetial equations with random parameters; Monte Carlo Method; collocation method; stochastic Gallerkin method; solution spaces of problems with random data; Karhunen-Loeve expansion; Mercer’s lemma; covariance matrix decomposition; convergence with respect to random variables; Bayesian methods; inverse analysis.

Requirements:

Attending the course. At most three absent classes are allowed.

Completing homeworks and individual seminar projects.

Syllabus of lectures:

1. Numerical solution of eliptic differential equations by finite element method. Finite difference method.

2. Random variable. Basic types.

3. Problems with uncertain data and solution of them.

4. Monte Carlo method.

5. Random field. Karhunen-Loeve expansion and Mercer lemma.

6. Colocation method.

7. Stochastic Galerkin method.

8. Approximation function spaces.

9. Solvers of linear systems.

10. Preconditioning.

11. Bayes methods.

Syllabus of tutorials:

1. Basic introduction to Matlab or to another solution software.

2.-3. Coding the finite element method for a second order problem in 1D.

4. Monte Carlo method.

5. Random field expansion.

6.-8. Numerical soltion of a second order problem in 1D with uncertain data.

9.-10. Stochastic Galerkin method.

11.-12. Consulting of individual tasks.

Study Objective:

Improving the students' skill in solving differential equations with uncertain data.

Study materials:

Gabriel J. Lord, Catherine E. Powell, Tony Shardlow: An Introduction to Computational Stochastic PDEs (Cambridge Texts in Applied Mathematics) 2014, Chapters 1 and 2 only.

Tarantola, A.: Inverse Problem Theory and Model Parameter Estimation, SIAM, 2005.

Xiu, D., Numerical Methods for Stochastic Computations, Princeton University Press, Princeton, New Jersey, 2010.

Ghanem, R. G., and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Springer Verlag, New York, 1991.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans: