News (updated 21.3.)

• The make-up oral exam will be held on Thursday April 20, 2023. The participants have been contacted via email.

Dates

General Information

Description

High-dimensional numerical integration plays a central role in contemporary study of uncertainty quantification. The analysis of how uncertainties associated with material parameters or the measurement configuration propagate within mathematical models leads to challenging high-dimensional integration problems, fueling the need to develop efficient numerical methods for this task.

Modern quasi-Monte Carlo (QMC) methods are based on tailoring specially designed cubature rules for high-dimensional integration problems. By leveraging the smoothness and anisotropy of an integrand, it is possible to achieve faster-than-Monte Carlo convergence rates. QMC methods have become a popular tool for solving partial differential equations (PDEs) involving random coefficients, a central topic within the field of uncertainty quantification.

This course provides an introduction to uncertainty quantification and how QMC methods can be applied to solve problems arising within this field.

Target audience

The course is intended for mathematics students at the Master's level and above.

Prerequisites

Multivariable calculus, linear algebra, basic probability theory, and MATLAB (or some other programming language).

Completing the course

The conditions for completing this course are successfully completing and submitting at least 60% of the course's exercises and successfully passing the course exam.

Registration

• Please register to the course via Campus Management (CM), then you will be automatically registered in MyCampus/Whiteboard as well. Please note the deadlines indicated there. For further information and in case of any problems, please consult the Campus Management's Help for Students.
• Non-FU students should register to the course in KVV (Whiteboard)

Lecture notes

Lecture notes will be published here after each week's lecture.

Week 1: Hilbert spaces, Hilbert projection theorem, orthogonal decomposition
Week 2: Dual space, Riesz representation theorem, adjoint operator, Lax–Milgram lemma
Week 3: Fourier transform, Sobolev spaces
Week 4: Lipschitz domain, Trace theorem, elliptic PDE
Week 5: Galerkin method, Céa's lemma, finite element programming (files: pde_ex.m, FEMdata.m, UpdateStiffness.m)
Week 6: Brief overview of probability theory, Karhunen–Loève expansion, elliptic PDEs with random coefficients (files: lognormal_demo.m)
Week 7: Quasi-Monte Carlo methods (finally!), reproducing kernel Hilbert space (RKHS), worst-case error (files: lognormal_demo2.m / note about the implementation)
Week 8: Randomly shifted rank-1 lattice rules, shift-averaged worst-case error, component-by-component (CBC) construction
Week 9: CBC error bound
Week 10: Implementing CBC and the fast CBC algorithm (files: recording, fastcbc.m, generator.m)
Week 11: Application of QMC to elliptic PDEs endowed with a uniform and affine random diffusion coefficient
Weeks 12-13: Dimension truncation, finite element, and overall error analysis for the uniform and affine model
Week 14: Application of QMC to elliptic PDEs endowed with a lognormal random diffusion coefficient
Week 15: Summary

Exercise sheets

Weekly exercises will be published here after each lecture.

Exercise 1
Exercise 2
Exercise 3 (note about the first problem)
Exercise 4 (note about the fourth problem)
Exercise 5 (files: week5.mat, model solutions: ex3.m, ex4.m / ex3.py, ex4.py)
Exercise 6 (files: pardemo.m / pardemo.py, model solutions: w6e2.m, w6e3.m / w6e2.py, w6e3.py, note about the integral appearing in task 2, note about least squares regression in task 2)
Exercise 7 (files: week7.mat, offtheshelf.txt, helpful programs: parsumdemo.py, demo1.py, demo2.py / demo1.m, demo2.m, model solutions: w7e1.m, w7e2.m, w7e3.m, w7e4.m / w7e1.py, w7e2.py, w7e3.py, w7e4.py)
Exercise 8
Exercise 9 (erratum)
Exercise 10 
Exercise 11 (helpful programs: tut1.m, tut2.m, tut3.m, tut4.m / tut1.py, tut2.py, tut3.py, tut4.py; files: FEM1.mat, FEM2.mat, FEM3.mat, FEM4.mat, FEM5.mat, offtheshelf2048.txt; model solutions: w11e1.m, w11e2.m, w11e3.m, w11e4.m / w11e1.py, w11e2.py, w11e3.py, w11e4.py)
Bonus exercises

Please note that the bonus exercises will not be graded and do not need to be returned.

Contact

Literature

The following books will be relevant:

• O. P. Le Maître and O. M. Knio. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York, 2010.
• R. C. Smith. Uncertainty Quantification: Theory, Implementation, and Applications, volume 12 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.
• T. J. Sullivan. Introduction to Uncertainty Quantification. Springer, New York, in press.
• D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton, NJ, 2010.