Uncertainty Quantification and Quasi-Monte Carlo
News
- There will be no exercise session on Tuesday 13 May.
- All lectures and exercises will now take place in room A6/032.
Dates
Lectures | Mon 10:15-11:45 | A6/032 | Dr. Vesa Kaarnioja |
Exercises | Tue 10:15-11:45 | A6/032 | Dr. Vesa Kaarnioja |
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 Python (or some other programming language).
Completing the course
To be eligible to sit for the final exam, a student must earn a minimum of cumulative 60% points from the exercises. The course evaluation is based on the final 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 before each week's lecture.
- Weeks 1-2: Hilbert spaces, Sobolev spaces, elliptic partial differential equations (PDEs)
- Week 3: Galerkin method, Céa's lemma, finite element programming (files: fem.py)
- Week 4: Brief overview of probability theory, Karhunen–Loève expansion, elliptic PDEs with random coefficients (files: lognormal_demo.py / note about least squares regression)
- Weeks 5-6 (12.5. and 19.5.2025): Quasi-Monte Carlo methods, randomly shifted rank-1 lattice rules, component-by-component (CBC) construction (files: lognormal_demo2.py / note about the implementation)
- Week 7 (26.5.2025): CBC error bound
- Week 8 (2.6.2025): Implementing the CBC and the fast CBC algorithm (files: fastcbc.py)
- Week 9 (16.6.2025): Application of QMC to elliptic PDEs endowed with a uniform and affine random diffusion coefficient
- Week 10 (23.6.2025): Dimension truncation, finite element, and overall error analysis for the uniform and affine model
- Week 11 (30.6.2025): Application of QMC to elliptic PDEs endowed with a lognormal random diffusion coefficient
Exercise sheets
Exercises will be published here regularly.
- Exercise 1
- Exercise 2 (files: femdata.mat, due by: 6 May, 10:15 am)
- Exercise 3 (extended deadline: 20 May, 10:15 am)
Contact
Dr. Vesa Kaarnioja | vesa.kaarnioja@fu-berlin.de | Arnimallee 6, room 212 Consulting hours: By appointment |