Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods

Scarabosio, Laura; Wohlmuth, Barbara; Oden, J. T.; Faghihi, Danial

doi:10.1016/j.camwa.2019.04.014

Cited by 4 publications

(1 citation statement)

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“…[61], (iii) they based their error control on the MSE error and did not separate the statistical error and bias errors as we did. On a less related note, Scarabosio et al [85] used goal-oriented model adaptivity based on a hierarchical control variate using two and three levels. They based their error estimates on verifying the current model output pathwise against a higher-fidelity model.…”

Section: Introductionmentioning

confidence: 99%

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

Beck¹,

Liu²,

Schwerin³

et al. 2022

Preprint

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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 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), pp. 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 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

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Section: Introductionmentioning

confidence: 99%