Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates

TL;DR

This paper introduces an adaptive multilevel Monte Carlo algorithm tailored for elliptic PDEs with lognormal coefficients and geometric singularities, improving computational efficiency over standard methods.

Contribution

The paper develops a novel AMLMC algorithm using adaptive meshes and dual residual error estimates, specifically designed for PDEs with geometric singularities and lognormal coefficients.

Findings

Enhanced efficiency over standard Monte Carlo methods.

Effective handling of geometric singularities in PDE solutions.

Demonstrated improvements through numerical experiments.

Abstract

We present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of $\mathbb{R}^d$. Our AMLMC algorithm is built on the results of the weak convergence rates in the work [Moon et al., BIT Numer. Math., 46 (2006), 367-407] for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. For a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. We discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation. From the numerical experiments, based on a Fourier expansion of the coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo and standard MLMC for a problem with a singularity similar to that at the tip of a slit modeling a crack.