Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients

Barajas-Solano, David A.; Tartakovsky, Daniel M.

doi:10.1137/130930108

Cited by 27 publications

(20 citation statements)

References 37 publications

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“…Since realistic random field models often need a rather large number of stochastic degrees of freedom (>100 s) for their accurate representation of the heterogeneous porous media property, the main computational challenge is to efficiently solve these stochastic PDEs. Several stochastic techniques like perturbation/moment methods [ Zhang , ], stochastic Galerkin methods [ Ghanem , ], stochastic collocation methods [ Nobile et al ., , ; Barajas‐Solano and Tartakovsky , ], and model order reduction methods [ Pasetto et al ., ] have been applied to porous media flow problems [ Ghanem , ; Zhang and Lu , ; Li and Zhang , ; Laloy et al ., ; Zhang et al ., ]. Among these techniques, the moment methods have wide application for their easy implementation.…”

Section: Introductionmentioning

confidence: 99%

An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations

Zhang

Webster

et al. 2016

Water Resources Research

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In this work, we develop an improved multilevel Monte Carlo (MLMC) method for estimating cumulative distribution functions (CDFs) of a quantity of interest, coming from numerical approximation of large‐scale stochastic subsurface simulations. Compared with Monte Carlo (MC) methods, that require a significantly large number of high‐fidelity model executions to achieve a prescribed accuracy when computing statistical expectations, MLMC methods were originally proposed to significantly reduce the computational cost with the use of multifidelity approximations. The improved performance of the MLMC methods depends strongly on the decay of the variance of the integrand as the level increases. However, the main challenge in estimating CDFs is that the integrand is a discontinuous indicator function whose variance decays slowly. To address this difficult task, we approximate the integrand using a smoothing function that accelerates the decay of the variance. In addition, we design a novel a posteriori optimization strategy to calibrate the smoothing function, so as to balance the computational gain and the approximation error. The combined proposed techniques are integrated into a very general and practical algorithm that can be applied to a wide range of subsurface problems for high‐dimensional uncertainty quantification, such as a fine‐grid oil reservoir model considered in this effort. The numerical results reveal that with the use of the calibrated smoothing function, the improved MLMC technique significantly reduces the computational complexity compared to the standard MC approach. Finally, we discuss several factors that affect the performance of the MLMC method and provide guidance for effective and efficient usage in practice.

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Section: Introductionmentioning

confidence: 99%

An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations

Zhang

Webster

et al. 2016

Water Resources Research

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“…As a result, they all suffer from the so-called "curse of dimensionality": their computational cost increases exponentially with the number of stochastic dimensions. The state-of-the-art PC methods, including enhancements like ANOVA, become less computationally efficient than MCS for approximately 100 random dimensions [9]- [11]. Based on available data, we estimate that the heterogeneity and multiscale dynamics of the Hanford Site requires at least 1,000 random dimensions for adequate representation, which is out of reach of the PC methods.…”

Section: State Of the Art A State-of-the-art In Spde Solversmentioning

confidence: 99%

Highly-scalable, Physics-Informed GANs for Learning Solutions of Stochastic PDEs

Liu

Prabhat

Karniadakis

et al. 2019

2019 IEEE/ACM Third Workshop on Deep Learning on Supercomputers (DLS)

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Uncertainty quantification for forward and inverse problems is a central challenge across physical and biomedical disciplines. We address this challenge for the problem of modeling subsurface flow at the Hanford Site by combining stochastic computational models with observational data using physics-informed GAN models. The geographic extent, spatial heterogeneity, and multiple correlation length scales of the Hanford Site require training a computationally intensive GAN model to thousands of dimensions. We develop a hierarchical scheme for exploiting domain parallelism, map discriminators and generators to multiple GPUs, and employ efficient communication schemes to ensure training stability and convergence. We developed a highly optimized implementation of this scheme that scales to 27,500 NVIDIA Volta GPUs and 4584 nodes on the Summit supercomputer with a 93.1% scaling efficiency, achieving peak and sustained half-precision rates of 1228 PF/s and 1207 PF/s.

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“…While the SC might underperform Monte Carlo when the hydraulic properties are spatially varying random functions with relatively short correlation lengths and/or relatively high variances (Barajas‐Solano & Tartakovsky, ), it is well suited for the low‐dimensional probability spaces, such as

N_{par} = 6

considered in the present study (e.g., Ciriello et al, , and references therein).…”

Section: Probabilistic Framework For Subsurface Contamination Assessmentmentioning

confidence: 99%

Impact of Hydrogeological Uncertainty on Estimation of Environmental Risks Posed by Hydrocarbon Transportation Networks

Ciriello

Lauriola

Bonvicini

et al. 2017

Water Resources Research

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Ubiquitous hydrogeological uncertainty undermines the veracity of quantitative predictions of soil and groundwater contamination due to accidental hydrocarbon spills from onshore pipelines. Such predictions, therefore, must be accompanied by quantification of predictive uncertainty, especially when they are used for environmental risk assessment. We quantify the impact of parametric uncertainty on quantitative forecasting of temporal evolution of two key risk indices, volumes of unsaturated and saturated soil contaminated by a surface spill of light nonaqueous‐phase liquids. This is accomplished by treating the relevant uncertain parameters as random variables and deploying two alternative probabilistic models to estimate their effect on predictive uncertainty. A physics‐based model is solved with a stochastic collocation method and is supplemented by a global sensitivity analysis. A second model represents the quantities of interest as polynomials of random inputs and has a virtually negligible computational cost, which enables one to explore any number of risk‐related contamination scenarios. For a typical oil‐spill scenario, our method can be used to identify key flow and transport parameters affecting the risk indices, to elucidate texture‐dependent behavior of different soils, and to evaluate, with a degree of confidence specified by the decision‐maker, the extent of contamination and the correspondent remediation costs.

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Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients

Cited by 27 publications

References 37 publications

An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations

An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations

Highly-scalable, Physics-Informed GANs for Learning Solutions of Stochastic PDEs

Impact of Hydrogeological Uncertainty on Estimation of Environmental Risks Posed by Hydrocarbon Transportation Networks

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