scholarly journals Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media

2017 ◽  
Vol 4 (8) ◽  
pp. 170203 ◽  
Author(s):  
D. Crevillén-García ◽  
H. Power
Keyword(s):  
Porous Media ◽  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Input Parameter ◽  
Computational Cost ◽  
Gaussian Random Field ◽  
Heterogeneous Porous Media ◽  
Quasi Monte Carlo ◽  
Average Travel Time ◽  
The Mean

In this study, we apply four Monte Carlo simulation methods, namely, Monte Carlo, quasi-Monte Carlo, multilevel Monte Carlo and multilevel quasi-Monte Carlo to the problem of uncertainty quantification in the estimation of the average travel time during the transport of particles through random heterogeneous porous media. We apply the four methodologies to a model problem where the only input parameter, the hydraulic conductivity, is modelled as a log-Gaussian random field by using direct Karhunen–Loéve decompositions. The random terms in such expansions represent the coefficients in the equations. Numerical calculations demonstrating the effectiveness of each of the methods are presented. A comparison of the computational cost incurred by each of the methods for three different tolerances is provided. The accuracy of the approaches is quantified via the mean square error.

scholarly journals Multilevel and Quasi Monte Carlo methods for the calculation of the Expected Value of Partial Perfect Information

2021 ◽  
Author(s):  
Wei Fang ◽  
Zhenru Wang ◽  
Mike B Giles ◽  
Christopher H Jackson ◽  
Nicky J Welton ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Input Parameter ◽  
Computational Cost ◽  
Expected Value ◽  
Perfect Information ◽  
Parameter Estimates ◽  
Tree Model ◽  
Generalised Additive Model ◽  
Approximation Techniques ◽  
Quasi Monte Carlo

The expected value of partial perfect information (EVPPI) provides an upper bound on the value of collecting further evidence on a set of inputs to a cost-effectiveness decision model. Standard Monte Carlo (MC) estimation of EVPPI is computationally expensive as it requires nested simulation. Alternatives based on regression approximations to the model have been developed, but are not practicable when the number of uncertain parameters of interest is large and when parameter estimates are highly correlated. The error associated with the regression approximation is difficult to determine, while MC allows the bias and precision to be controlled. In this paper, we explore the potential of Quasi Monte-Carlo (QMC) and Multilevel Monte-Carlo (MLMC) estimation to reduce computational cost of estimating EVPPI by reducing the variance compared with MC, while preserving accuracy. In this paper, we develop methods to apply QMC and MLMC to EVPPI, addressing particular challenges that arise where Markov Chain Monte Carlo (MCMC) has been used to estimate input parameter distributions. We illustrate the methods using a two examples: a simplified decision tree model for treatments for depression, and a complex Markov model for treatments to prevent stroke in atrial fibrillation, both of which use MCMC inputs. We compare the performance of QMC and MLMC with MC and the approximation techniques of Generalised Additive Model regression (GAM), Gaussian process regression (GP), and Integrated Nested Laplace Approximations (INLA-GP). We found QMC and MLMC to offer substantial computational savings when parameter sets are large and correlated, and when the EVPPI is large. We also find GP and INLA-GP to be biased in those situations, while GAM cannot estimate EVPPI for large parameter sets.

scholarly journals Multilevel and Quasi Monte Carlo Methods for the Calculation of the Expected Value of Partial Perfect Information

2021 ◽  
pp. 0272989X2110263
Author(s):  
Wei Fang ◽  
Zhenru Wang ◽  
Michael B. Giles ◽  
Chris H. Jackson ◽  
Nicky J. Welton ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Input Parameter ◽  
Computational Cost ◽  
Additive Model ◽  
Expected Value ◽  
Perfect Information ◽  
Parameter Estimates ◽  
Tree Model ◽  
Approximation Techniques ◽  
Quasi Monte Carlo

The expected value of partial perfect information (EVPPI) provides an upper bound on the value of collecting further evidence on a set of inputs to a cost-effectiveness decision model. Standard Monte Carlo estimation of EVPPI is computationally expensive as it requires nested simulation. Alternatives based on regression approximations to the model have been developed but are not practicable when the number of uncertain parameters of interest is large and when parameter estimates are highly correlated. The error associated with the regression approximation is difficult to determine, while MC allows the bias and precision to be controlled. In this article, we explore the potential of quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) estimation to reduce the computational cost of estimating EVPPI by reducing the variance compared with MC while preserving accuracy. We also develop methods to apply QMC and MLMC to EVPPI, addressing particular challenges that arise where Markov chain Monte Carlo (MCMC) has been used to estimate input parameter distributions. We illustrate the methods using 2 examples: a simplified decision tree model for treatments for depression and a complex Markov model for treatments to prevent stroke in atrial fibrillation, both of which use MCMC inputs. We compare the performance of QMC and MLMC with MC and the approximation techniques of generalized additive model (GAM) regression, Gaussian process (GP) regression, and integrated nested Laplace approximations (INLA-GP). We found QMC and MLMC to offer substantial computational savings when parameter sets are large and correlated and when the EVPPI is large. We also found that GP and INLA-GP were biased in those situations, whereas GAM cannot estimate EVPPI for large parameter sets.

Parametric uncertainty quantification in the Rothermel model with randomised quasi-Monte Carlo methods

2015 ◽  
Vol 24 (3) ◽  
pp. 307 ◽  
Author(s):  
Yaning Liu ◽  
Edwin Jimenez ◽  
M. Yousuff Hussaini ◽  
Giray Ökten ◽  
Scott Goodrick
Keyword(s):  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Monte Carlo Methods ◽  
Wildland Fire ◽  
Parametric Uncertainty ◽  
Monte Carlo Sampling ◽  
Model Parameters ◽  
Quasi Monte Carlo ◽  
Control Variate ◽  
Fuel Model

Rothermel's wildland surface fire model is a popular model used in wildland fire management. The original model has a large number of parameters, making uncertainty quantification challenging. In this paper, we use variance-based global sensitivity analysis to reduce the number of model parameters, and apply randomised quasi-Monte Carlo methods to quantify parametric uncertainties for the reduced model. The Monte Carlo estimator used in these calculations is based on a control variate approach applied to the sensitivity derivative enhanced sampling. The chaparral fuel model, selected from Rothermel's 11 original fuel models, is studied as an example. We obtain numerical results that improve the crude Monte Carlo sampling by factors as high as three orders of magnitude.

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

2019 ◽  
Vol 53 (5) ◽  
pp. 1507-1552 ◽  
Author(s):  
L. Herrmann ◽  
C. Schwab
Keyword(s):  
Monte Carlo ◽  
Random Field ◽  
Gaussian Random Field ◽  
Function Spaces ◽  
Monte Carlo Integration ◽  
Superior Performance ◽  
Elliptic Pdes ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Quasi Monte Carlo

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

scholarly journals Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

SPE Journal ◽  
2016 ◽  
Vol 21 (01) ◽  
pp. 144-151 ◽  
Author(s):  
Mehdi Ghommem ◽  
Eduardo Gildin ◽  
Mohammadreza Ghasemi
Keyword(s):  
Porous Media ◽  
Model Reduction ◽  
Flow Behavior ◽  
Computational Cost ◽  
Heterogeneous Porous Media ◽  
Production Optimization ◽  
Dynamic Mode ◽  
Two Phase ◽  
Fine Grid ◽  
Mode Decomposition

Summary In this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather post-processes a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their proper-orthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobility-related term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization.

scholarly journals Modeling of 3D Rock Porous Media by Combining X-Ray CT and Markov Chain Monte Carlo

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Wei Lin ◽  
Xizhe Li ◽  
Zhengming Yang ◽  
Shengchun Xiong ◽  
Yutian Luo ◽  
...  
Keyword(s):  
Porous Media ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Heterogeneous Porous Media ◽  
Pore Throat ◽  
Tight Sandstone ◽  
Sandstone Sample ◽  
Digital Rock ◽  
Multi Scale

Abstract Rocks contain multi-scale pore structures, with dimensions ranging from nano- to sample-scale, the inherent tradeoff between imaging resolution and sample size limits the simultaneous characterization of macro-pores and micro-pores using single-resolution imaging. Here, we developed a new hybrid digital rock modeling approach to cope with this open challenge. We first used micron-CT to construct the 3D macro-pore digital rock of tight sandstone, then performed high-resolution SEM on the three orthogonal surfaces of sandstone sample, thus reconstructed the 3D micro-pore digital rock by Markov chain Monte Carlo (MCMC) method; finally, we superimposed the macro-pore and micro-pore digital rocks to achieve the integrated digital rock. Maximal ball algorithm was used to extract pore-network parameters of digital rocks, and numerical simulations were completed with Lattice-Boltzmann method (LBM). The results indicate that the integrated digital rock has anisotropy and good connectivity comparable with the real rock, and porosity, pore-throat parameters and intrinsic permeability from simulations agree well with the values acquired from experiments. In addition, the proposed approach improves the accuracy and scale of digital rock modeling and can deal with heterogeneous porous media with multi-scale pore-throat system.

Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties

2018 ◽  
Author(s):  
Alen Alexanderian ◽  
William Reese ◽  
Ralph C. Smith ◽  
Meilin Yu
Keyword(s):  
Porous Media ◽  
Random Field ◽  
Material Properties ◽  
Pressure Field ◽  
Input Parameter ◽  
Gaussian Random Field ◽  
Porous Media Flow ◽  
Expansion Technique ◽  
Low Dimensional ◽  
Permeability Field

We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous media flow. The covariance matrix of pressure is constructed via Monte Carlo sampling. The truncated Karhunen–Loève (KL) expansion technique is used to decompose the log-permeability field, as well as the random pressure field resulting from random permeability. We find that although very high-dimensional representation is needed to recover the permeability field when the correlation length is small, the pressure field is not sensitive to high-oder KL terms of input parameter, and itself can be modeled using a low-dimensional model. Thus a low-rank representation of the pressure field in a low-dimensional parameter space is constructed using the truncated KL expansion technique.

Sign in / Sign up

Export Citation Format

Share Document