Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

Barker, Andrew T.; Lee, Chak Shing; Vassilevski, Panayot S.

doi:10.1137/16m1077581

Cited by 11 publications

(6 citation statements)

References 45 publications

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“…In this case, we have η w H h (cf., [5]) where H stands for the diameter of the aggregates. This fact, combined with a simple argument relating the right hand side of the discrete problem, f , and the L 2 -norm f 0 of the right hand side function f (as shown in [27]), we conclude that the first term η w D − 1 2 Au H f 0 . If we want to balance the second term with the first one, we need to choose 2q ν H (assume Chebyshev polynomial or CG used).…”

Section: Example: Linear Finite Elements For Laplace Equationsupporting

confidence: 58%

“…In the PDE case this challenge seems resolvable if large enough coarsening factor (H/h) is employed, whereas in the graph application for graphs with irregular degree distribution, in addition to high coarsening factor one may need to employ graph disaggregation (cf., [16]), which is left for a possible future study. Additionally, in the PDE case, it is of interest to extend the present results to other types of PDEs such as ones posed in H(curl) and H(div), which will provide alternatives to the existing AMGe upscaling methods (cf., [17], [13], and [1]).…”

Section: Remarks For Elliptic Problems With High Contrast Coefficientsmentioning

confidence: 99%

“…We present numerical results illustrating the effectiveness of the proposed method. We would like to point out that other computationally feasible AMG-type upscaling approaches were presented in [1] and [13] for problems that can be formulated in a mixed (saddle-point) form.…”

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confidence: 99%

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Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Hu¹,

Vassilevski²

2019

SIAM J. Matrix Anal. Appl.

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Algebraic multigrid (AMG) coarse spaces are commonly constructed so that they exhibit the so-called weak approximation (WAP) property which is necessary and sufficient condition for uniform two-grid convergence. This paper studies a modification of such coarse spaces so that the modified ones provide approximation in energy norm. Our modification is based on the projection in energy norm onto an orthogonal complement of original coarse space. This generally leads to dense modified coarse space matrices which is hence computationally infeasible. To remedy this, based on the fact that the projection involves inverse of a well-conditioned matrix, we use polynomials to approximate the projection and, therefore, obtain a practical, sparse modified coarse matrix and prove that the modified coarse space maintains computationally feasible approximation in energy norm. We present some numerical results for both, PDE discretization matrices as well as graph Laplacian ones, which are in accordance with our theoretical results.

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Section: Example: Linear Finite Elements For Laplace Equationsupporting

confidence: 58%

Section: Remarks For Elliptic Problems With High Contrast Coefficientsmentioning

confidence: 99%

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Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Hu¹,

Vassilevski²

2019

SIAM J. Matrix Anal. Appl.

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“…Other works that focus on fractured media include Long [40], Pouya and Fouché [43], and Lang et al [39]. Barker et al [65] consider the spectral graph properties of networks and developed a reservoir simulation application involving a Finite Volume discretization of a 3D heterogeneous continuum, re-interpreted as a 3D network of links.…”

Section: Introductionmentioning

confidence: 99%

Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses)

Renard

Ababou

2022

Geosciences

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When conducting numerical upscaling, either for a fractured or a porous medium, it is important to account for anisotropy because in general, the resulting upscaled conductivity is anisotropic. Measurements made at different scales also demonstrate the existence of anisotropy of hydraulic conductivity. At the “microscopic” scale, the anisotropy results from the preferential flatness of grains, presence of shale, or variation of grain size in successive laminations. At a larger scale, the anisotropy results from preferential orientation of highly conductive geological features (channels, fracture families) or alternations of high and low conductive features (stratification, bedding, crossbedding). Previous surveys of homogenization techniques demonstrate that a wide variety of approaches exists to define and calculate the equivalent conductivity tensor. Consequently, the resulting equivalent conductivities obtained by these different methods are not necessarily equal, and they do not have the same mathematical properties (some are symmetric, others are not, for example). We present an overview of different techniques allowing a quantitative evaluation of the anisotropic equivalent conductivity for heterogeneous porous media, via numerical simulations and, in some cases, analytical approaches. New approaches to equivalent permeability are proposed for heterogeneous media, as well as discontinuous (composite) media, and also some extensions to 2D fractured networks. One of the main focuses of the paper is to explore the relations between these various definitions and the resulting properties of the anisotropic equivalent conductivity, such as tensorial or non-tensorial behavior of the anisotropic conductivity; symmetry and positiveness of the conductivity tensor (or not); dual conductivity/resistivity tensors; continuity and robustness of equivalent conductivity with respect to domain geometry and boundary conditions. In this paper, we emphasize some of the implications of the different approaches for the resulting equivalent permeabilities.

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“…In particular, the solver can also be applied to solve coarse saddle point problems coming from coarsening of the graph Laplacian problems, which has application in upscaling of finite volume discretization in reservoir simulations, see [6] for a detail discussion.…”

Section: Efficient Solver For Coarse Saddle Point Systemsmentioning

confidence: 99%

Generalization of mixed multiscale finite element methods with applications

Lee¹

2016

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Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms.

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Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

Cited by 11 publications

References 45 publications

Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses)

Generalization of mixed multiscale finite element methods with applications

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