Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

Abstract
In this study, 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 Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work (Moon et al., 2006) 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 above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, 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. Furthermore, 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 diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a crack.

Citation
Beck, J., Liu, Y., von Schwerin, E., & Tempone, R. (2022). Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates. Computer Methods in Applied Mechanics and Engineering, 115582. https://doi.org/10.1016/j.cma.2022.115582

Acknowledgements
This publication is based on work supported by the Alexander von Humboldt Foundation and the King Abdullah University of Science and Technology (KAUST) office of sponsored research (OSR) under Award No. OSR-2019-CRG8-4033. The authors thank Daniele Boffi and Alexander Litvinenko for fruitful discussions. We also acknowledge the use of the following open-source software packages: deal.II [72], PAPI [71].

Publisher
Elsevier BV

Journal
Computer Methods in Applied Mechanics and Engineering

DOI
10.1016/j.cma.2022.115582

arXiv
2206.10314

Additional Links
https://linkinghub.elsevier.com/retrieve/pii/S0045782522005552

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