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

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    2206.10314.pdf
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    Type
    Preprint
    Authors
    Beck, Joakim cc
    Liu, Yang cc
    von Schwerin, Erik cc
    Tempone, Raul cc
    KAUST Department
    KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    KAUST Grant Number
    OSR-2019- CRG8-4033
    Date
    2022-06-21
    Permanent link to this record
    http://hdl.handle.net/10754/679297
    
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    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 Rd. 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.
    Sponsors
    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 [4], PAPI [90]
    Publisher
    arXiv
    arXiv
    2206.10314
    Additional Links
    https://arxiv.org/pdf/2206.10314.pdf
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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