Estimation of distributions via multilevel Monte Carlo with stratified sampling

Taverniers, Søren; Tartakovsky, Daniel M.

doi:10.1016/j.jcp.2020.109572

Cited by 23 publications

(10 citation statements)

References 32 publications

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“…The need to improve this slow convergence rate drives the field of UQ. Multilevel/multifidelity Monte Carlo can speedup standard Monte Carlo by orders of magnitude (e.g., Berrone et al., 2020; Berrone, Borio, et al., 2018; Berrone, Canuto, et al., 2018, O’Malley et al., 2018), although such a performance is not guaranteed if the goal is to compute the full distribution of a quantity of interest rather than its mean and variance (Taverniers & Tartakovsky, 2020; Taverniers et al., 2020).…”

Section: Uncertainty Quantification Multifidelity Models and Emulatorsmentioning

confidence: 99%

From Fluid Flow to Coupled Processes in Fractured Rock: Recent Advances and New Frontiers

Viswanathan

Ajo‐Franklin

Birkhölzer

et al. 2022

Reviews of Geophysics

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113

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Quantitative predictions of natural and induced phenomena in fractured rock is one of the great challenges in the Earth and Energy Sciences with far‐reaching economic and environmental impacts. Fractures occupy a very small volume of a subsurface formation but often dominate fluid flow, solute transport and mechanical deformation behavior. They play a central role in CO2 sequestration, nuclear waste disposal, hydrogen storage, geothermal energy production, nuclear nonproliferation, and hydrocarbon extraction. These applications require predictions of fracture‐dependent quantities of interest such as CO2 leakage rate, hydrocarbon production, radionuclide plume migration, and seismicity; to be useful, these predictions must account for uncertainty inherent in subsurface systems. Here, we review recent advances in fractured rock research covering field‐ and laboratory‐scale experimentation, numerical simulations, and uncertainty quantification. We discuss how these have greatly improved the fundamental understanding of fractures and one's ability to predict flow and transport in fractured systems. Dedicated field sites provide quantitative measurements of fracture flow that can be used to identify dominant coupled processes and to validate models. Laboratory‐scale experiments fill critical knowledge gaps by providing direct observations and measurements of fracture geometry and flow under controlled conditions that cannot be obtained in the field. Physics‐based simulation of flow and transport provide a bridge in understanding between controlled simple laboratory experiments and the massively complex field‐scale fracture systems. Finally, we review the use of machine learning‐based emulators to rapidly investigate different fracture property scenarios and accelerate physics‐based models by orders of magnitude to enable uncertainty quantification and near real‐time analysis.

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Section: Uncertainty Quantification Multifidelity Models and Emulatorsmentioning

confidence: 99%

From Fluid Flow to Coupled Processes in Fractured Rock: Recent Advances and New Frontiers

Viswanathan

Ajo‐Franklin

Birkhölzer

et al. 2022

Reviews of Geophysics

Self Cite

113

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“…The rigorous homogenization in [39] enables the derivation of macroscopic quantities from microscale counterparts with clearly defined limits of applicability, in contrast to relying on phenomenological relations. While it may be possible to minimize the computational burden with an appropriately chosen simulation technique, such as a multilevel Monte Carlo method as in [40], the homogenization and solution of the corresponding closure equations are an integral feature of this multiscale physics-based model and thus the associated computational bottleneck is intrinsic. Moreover, the complicated dependencies among the components, some of which are viewed as CVs for the QoIs, demand specialized tools, such as PGMs, to recast the physics-based model into a probabilistic framework.…”

Section: Motivation For the Use Of Structured Probabilistic Modelsmentioning

confidence: 99%

“…We cast the (originally deterministic) homogenized problem into a probabilistic framework by modeling the CVs X c as random variables, see e.g. approach followed for a similar problem in [40]. The type and support of the distributions placed on X c need to reflect a combination of expert opinion, available data, physical and design constraints, and other domain knowledge, in order to ensure the generation of physically meaningful distributions on the QoIs Y .…”

Section: Choice Of Tunable Control Variables and Identification Of Co...mentioning

confidence: 99%

GINNs: Graph-Informed Neural Networks for multiscale physics

Hall

Taverniers

Katsoulakis

et al. 2021

Journal of Computational Physics

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“…The MLMC method was also used in [14] for nested conditional expectations from which the VaR and CVaR could be derived. An alternative smoothing of the characteristic function based on the Kernel Density Estimation (KDE) method was proposed in [32], combined with an MLMC estimator wherein stratification based sampling was applied at each level. The authors of [6] combined an approach to locate the discontinuity using a root-finding algorithm, followed by numerical pre-integration.…”

Section: Introductionmentioning

confidence: 99%

Quantifying uncertain system outputs via the multi-level Monte Carlo method -- distribution and robustness measures

Ayoul-Guilmard¹,

Ganesh²,

Krumscheid³

et al. 2022

Preprint

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In this work, we consider the problem of estimating the probability distribution, the quantile or the conditional expectation above the quantile, the so called Conditional-Valueat-Risk (CVaR), of output quantities of complex random differential models by the Multi-Level Monte Carlo (MLMC) method. We follow the approach of [22], which recasts the estimation of the above quantities to the computation of suitable parametric expectations. In this work, we present novel computable error estimators for the estimation of such quantities, which are then used to optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We demonstrate the efficiency and robustness of our adaptive continuation-MLMC in an array of numerical test cases.

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Estimation of distributions via multilevel Monte Carlo with stratified sampling

Cited by 23 publications

References 32 publications

From Fluid Flow to Coupled Processes in Fractured Rock: Recent Advances and New Frontiers

From Fluid Flow to Coupled Processes in Fractured Rock: Recent Advances and New Frontiers

GINNs: Graph-Informed Neural Networks for multiscale physics

Quantifying uncertain system outputs via the multi-level Monte Carlo method -- distribution and robustness measures

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