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

Location: seminar room 1.114 at Rudower Chaussee 25, 12489 Berlin (HU Berlin Campus Adlershof).

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