Abstract

Quasi-Monte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of high-dimensional integrands.
Also the convergence rate of essentially first order is superior to Monte Carlo sampling.
We study a class of integrands that arise as solutions of elliptic PDEs with log-Gaussian coefficients.
In particular, we focus on the overall computational cost of the algorithm.
We prove that certain multilevel QMC rules have a consistent accuracy and computational cost that is essentially of optimal order in terms of the degrees of freedom of the spatial Finite Element
discretization for a range of infinite-dimensional priors.
This is joint work with Christoph Schwab.

References:
[L. Herrmann, Ch. Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numer. Math. 141(1) pp. 63—102, 2019],
[L. Herrmann, Ch. Schwab: Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, to appear in ESAIM :M2AN],
[L. Herrmann: Strong convergence analysis of iterative solvers for random operator equations, SAM report, 2017-35, in review]