Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo

TL;DR

This paper introduces an adaptive multilevel quasi-Monte Carlo method for efficiently approximating quantities of interest in PDEs with lognormal coefficients, leveraging adaptive meshes, importance sampling, and control variates to reduce computational costs.

Contribution

It develops a novel adaptive multilevel QMC framework with variance reduction techniques for PDE-based uncertainty quantification involving high-dimensional lognormal parameters.

Findings

Achieves prescribed accuracy with lower computational cost than standard MLMC.

Effectively incorporates importance sampling and control variates.

Demonstrates efficiency of adaptive meshes in high-dimensional PDE problems.

Abstract

The efficient approximation of quantity of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient {parameterized by a 49-dimensional Gaussian random vector} and deterministic geometric singularities in bounded domains of $\mathbb{R}^d$. We analyze the parametric regularity and develop the multilevel implementation based on a sequence of adaptive meshes, developed in "Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates", \emph{CMAME}, 402 (2022), p. 115582. For further variance reduction, we incorporate importance sampling and introduce a level-0 control variate within the multilevel hierarchy. {Introducing such control variate can alter the optimal choice of initial mesh, further highlighting the advantages of adaptive meshes.} Numerical experiments demonstrate that our adaptive QMC algorithm achieves a prescribed accuracy at substantially lower computational cost than the standard multilevel Monte Carlo method.