Quasi-Monte Carlo methods for PDE uncertainty quantification
This minicourse is organized as part of the Winter School of SIAM/GAMM Student Chapter Berlin (21-22 February 2024, Institute of Mathematics, Humboldt-Universität zu Berlin).Dates
Lectures:
Wed Feb 21 2024 | 09:30-12:30 | Dr. Vesa Kaarnioja |
Thu Feb 22 2024 | 13:30-16:30 | Dr. Vesa Kaarnioja |
Contact: vesa.kaarnioja@fu-berlin.de
General information
Description
High-dimensional numerical integration plays a central role in the 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. Meanwhile, Bayesian inference can be used to express the solution to inverse problems in terms of a high-dimensional posterior distribution. Evaluating the mean or uncertainty of the posterior distribution involves the computation of high-dimensional integrals.
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 a brief introduction to forward and inverse uncertainty quantification for elliptic PDE problems using QMC methods.
Lecture notes
- Lecture notes (updated 22.2.)
- Programs: fastcbc.m, generator.m, affine_example.m, lognormal_example.m