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Uncertainty Quantification and Quasi-Monte Carlo (Wintersemester 2022/23)

News (updated 27.1.)

  • The 11th and final exercise sheet has been published (here). Please note that the deadline for returning your solutions is Tuesday, February 7, 12:15. We will have a tutorial during the exercise session on Tuesday, January 31, to help you get started with the exercises.
  • A "bonus" exercise sheet has been published (here). Please note that these exercises will not be graded and do not need to be returned.
  • Course oral exam: I have contacted the course participants via email in order to arrange individual oral exam appointments. If the proposed time is not suitable for you or if you have not received an oral exam appointment and wish to complete the course, please contact the lecturer.
  • We agreed during the first lecture that there will also be a live broadcast of future lectures online via Webex (note that the lectures will not be recorded). Webex ID: 2731 073 6893 (password: uqqmc). However, you are very welcome to continue attending the in-person lectures on Mondays in room SR009!
  • The exercise sessions will not be streamed via Webex.


Lectures Mon 12:15-14:00 A6/SR009 Dr. Vesa Kaarnioja
Exercises Tue 12:15-14:00 A3/SR115 Dr. Vesa Kaarnioja
Oral exam Mon February 13, 2023 A6/213  
Make-up oral exam TBA (late March or early April) TBA  

The course participants have been contacted via email to arrange individual oral exam appointments. If you have not been contacted and you wish to take part in the course exam, please contact the lecturer directly via email.

If you wish to register to the make-up oral exam, please send an email to the lecturer before March 20 to arrange an appointment.

General Information


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.


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.


  • 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: recordingfastcbc.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

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.pydemo1.pydemo2.py / demo1.mdemo2.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 (tutorial: January 31, helpful programs: tut1.m, tut2.m, tut3.m, tut4.m / tut1.py, tut2.py, tut3.py, tut4.py; deadline: February 7, 12:15, files: FEM1.mat, FEM2.mat, FEM3.mat, FEM4.mat, FEM5.mat, offtheshelf2048.txt)
Bonus exercises

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


Dr. Vesa Kaarnioja vesa.kaarnioja@fu-berlin.de Arnimallee 6, room 212
Consulting hours: By appointment