Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913975053058048 |
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| author | Beck, Joakim Liu, Yang von Schwerin, Erik Tempone, Raúl |
| author_facet | Beck, Joakim Liu, Yang von Schwerin, Erik Tempone, Raúl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo Beck, Joakim Liu, Yang von Schwerin, Erik Tempone, Raúl Numerical Analysis 65C05, 65N50, 65N22, 35R60 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. |
| title | Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo |
| topic | Numerical Analysis 65C05, 65N50, 65N22, 35R60 |
| url | https://arxiv.org/abs/2508.02925 |