An adaptive multiscale method for density-driven instabilities

Künze, Rouven; Lunati, Ivan

doi:10.1016/j.jcp.2012.02.025

Cited by 44 publications

(32 citation statements)

References 29 publications

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“…Although it offers an optimal dimensionality reduction with respect to the total mean squared error, the orthonormal basis might not be ideal to represent the information. The varimax algorithm [Kaiser, 1958] can be applied to find a suitable rotation that improve data interpretation while preserving the optimality of the result in terms of explained variance [Richman, 1986;Ramsay et al, 2009]. Therefore, without any further loss of information, the approximate curves can be written as…”

Section: Functional Reduction Of the Dimensionalitymentioning

confidence: 99%

Functional error modeling for uncertainty quantification in hydrogeology

Josset

Ginsbourger

Lunati

2015

Water Resources Research

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Approximate models (proxies) can be employed to reduce the computational costs of estimating uncertainty. The price to pay is that the approximations introduced by the proxy model can lead to a biased estimation. To avoid this problem and ensure a reliable uncertainty quantification, we propose to combine functional data analysis and machine learning to build error models that allow us to obtain an accurate prediction of the exact response without solving the exact model for all realizations. We build the relationship between proxy and exact model on a learning set of geostatistical realizations for which both exact and approximate solvers are run. Functional principal components analysis (FPCA) is used to investigate the variability in the two sets of curves and reduce the dimensionality of the problem while maximizing the retained information. Once obtained, the error model can be used to predict the exact response of any realization on the basis of the sole proxy response. This methodology is purpose-oriented as the error model is constructed directly for the quantity of interest, rather than for the state of the system. Also, the dimensionality reduction performed by FPCA allows a diagnostic of the quality of the error model to assess the informativeness of the learning set and the fidelity of the proxy to the exact model. The possibility of obtaining a prediction of the exact response for any newly generated realization suggests that the methodology can be effectively used beyond the context of uncertainty quantification, in particular for Bayesian inference and optimization.

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Section: Functional Reduction Of the Dimensionalitymentioning

confidence: 99%

Functional error modeling for uncertainty quantification in hydrogeology

Josset

Ginsbourger

Lunati

2015

Water Resources Research

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“…While still an open question in the computational hydrology community, recent theoretical (Battiato et al ; Battiato and Tartakovsky ) and computational (Boso and Battiato ) works suggest that a priori estimates of continuum scale quantities/parameters might be employed as adaptivity criteria. These include evaluation of gradients of continuum‐scale quantities (Battiato et al ; Kunze and Lunati ) and time‐ or space‐dependent macroscopic dimensionless numbers (Boso and Battiato ).…”

Section: Multiscale Analysis Platformmentioning

confidence: 99%

An Analysis Platform for Multiscale Hydrogeologic Modeling with Emphasis on Hybrid Multiscale Methods

et al. 2014

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One of the most significant challenges faced by hydrogeologic modelers is the disparity between the spatial and temporal scales at which fundamental flow, transport, and reaction processes can best be understood and quantified (e.g., microscopic to pore scales and seconds to days) and at which practical model predictions are needed (e.g., plume to aquifer scales and years to centuries). While the multiscale nature of hydrogeologic problems is widely recognized, technological limitations in computation and characterization restrict most practical modeling efforts to fairly coarse representations of heterogeneous properties and processes. For some modern problems, the necessary level of simplification is such that model parameters may lose physical meaning and model predictive ability is questionable for any conditions other than those to which the model was calibrated. Recently, there has been broad interest across a wide range of scientific and engineering disciplines in simulation approaches that more rigorously account for the multiscale nature of systems of interest. In this article, we review a number of such approaches and propose a classification scheme for defining different types of multiscale simulation methods and those classes of problems to which they are most applicable. Our classification scheme is presented in terms of a flowchart (Multiscale Analysis Platform), and defines several different motifs of multiscale simulation. Within each motif, the member methods are reviewed and example applications are discussed. We focus attention on hybrid multiscale methods, in which two or more models with different physics described at fundamentally different scales are directly coupled within a single simulation. Very recently these methods have begun to be applied to groundwater flow and transport simulations, and we discuss these applications in the context of our classification scheme. As computational and characterization capabilities continue to improve, we envision that hybrid multiscale modeling will become more common and also a viable alternative to conventional single-scale models in the near future.

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“…The breakthrough curves are obtained by solving the classical advection-dispersion equation in transient state using a finite volume technique [43,44]. The spatial discretization is kept identical to the one used for the geological simulations.…”

Section: Flow and Transport Simulationsmentioning

confidence: 99%

“…The first one, p 1 x ðtÞ, is based on simplified physics. We use the same solver [43,44] and the same spatial and temporal resolution as for the accurate model based on the full physics, but we disregard diffusion and dispersion effects. The numerical simulation thereby only accounts for advection and numerical dispersion phenomena.…”

Section: Two Different Proxiesmentioning

confidence: 99%

Distance-based kriging relying on proxy simulations for inverse conditioning

Ginsbourger

Rosspopoff

Pirot

et al. 2013

Advances in Water Resources

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a b s t r a c tLet us consider a large set of candidate parameter fields, such as hydraulic conductivity maps, on which we can run an accurate forward flow and transport simulation. We address the issue of rapidly identifying a subset of candidates whose response best match a reference response curve. In order to keep the number of calls to the accurate flow simulator computationally tractable, a recent distance-based approach relying on fast proxy simulations is revisited, and turned into a non-stationary kriging method where the covariance kernel is obtained by combining a classical kernel with the proxy. Once the accurate simulator has been run for an initial subset of parameter fields and a kriging metamodel has been inferred, the predictive distributions of misfits for the remaining parameter fields can be used as a guide to select candidate parameter fields in a sequential way. The proposed algorithm, Proxy-based Kriging for Sequential Inversion (ProKSI), relies on a variant of the Expected Improvement, a popular criterion for kriging-based global optimization. A statistical benchmark of ProKSI's performances illustrates the efficiency and the robustness of the approach when using different kinds of proxies.

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An adaptive multiscale method for density-driven instabilities

Cited by 44 publications

References 29 publications

Functional error modeling for uncertainty quantification in hydrogeology

Functional error modeling for uncertainty quantification in hydrogeology

An Analysis Platform for Multiscale Hydrogeologic Modeling with Emphasis on Hybrid Multiscale Methods

Distance-based kriging relying on proxy simulations for inverse conditioning

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