On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

Kumar, Prashant; Rodrigo, Carmen; Gaspar, Francisco José; Oosterlee, Cornelis W.

doi:10.1137/18m1173769

Cited by 21 publications

(22 citation statements)

References 53 publications

(74 reference statements)

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“…Numerical results for various other parameter settings can be found online at https://people.cs.kuleuven.be/~pieterjan.robbe/copper2019. Finally, we refer to Kumar et al 17 for a local Fourier analysis (LFA) of the model problem with isotropic Matérn covariance.…”

Section: Multiple Semicoarsened Multigridmentioning

confidence: 99%

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

Robbe

Nuyens

Vandewalle

2020

Numerical Linear Algebra App

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In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncertain permeability of the material is modelled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [15,16,17], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method, that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC) [6] to Multi-Index Monte Carlo (MIMC) [9]. We present an unbiased MIMC method that reuses the MSG coarse solutions, similar to the work in [11]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and unifies the previous work on adaptive MIMC [19] and unbiased estimation [18]. We analyse the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

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Section: Multiple Semicoarsened Multigridmentioning

confidence: 99%

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

Robbe

Nuyens

Vandewalle

2020

Numerical Linear Algebra App

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“…Due to a small correlation length and low spatial regularity, the numerical solutions of the PDE with random coefficients based on Φ 2 are comparatively more expensive to compute than those obtained with Φ 1 . A comprehensive study on the computational cost of solving elliptic PDEs with different Matérn parameters can be found in [28]. We will study the effect of covariance functions on the performance of the solver for the Richards' equation.…”

Section: Sampling Of Non-gaussian Random Fieldsmentioning

confidence: 99%

“…Within each modified Picard iteration, a diffusion equation with variable coefficients needs to be solved. For this, we propose to utilize the cell-centered multigrid (CCMG) for heterogeneous diffusion coefficients, as proposed in [28,38,39]. The CCMG algorithm is efficient as it is constructed with a simple set of transfer operators and it has been demonstrated to perform well for a large class of highly heterogeneous and also jumping diffusion coefficients [28].…”

Section: Modified Picard Iteration Combined With the Cell-centered Mumentioning

confidence: 99%

“…In [28], the W (2, 2)-cycle was found to be a very robust and efficient multigrid cycling strategy, and, therefore, we also employ this cycle in our experiments. The number of multigrid iterations is based on the We consider the modified Picard method in this article as it is widely adopted, although many modifications have been proposed to improve its robustness.…”

Section: Cell-centered Multigridmentioning

confidence: 99%

“…We furthermore propose a solution method for Richards' equation based on a combination of the modified Picard method [2] and a cell-centered multigrid, as proposed in [28]. We benchmark the performance of this combined solver in a probabilistic framework.…”

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A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

et al. 2019

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We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that are modeled using the Richards' equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards' equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards' equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared to the original MC estimator.

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Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

de Oliveira,

Pinto,

Rodrigo

et al. 2023

Numerical Meth Engineering

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Two‐phase flow problems in porous media can be found in several areas, such as Geomechanics, Hydrogeology, Engineering and Biomedicine, for example. Typically, these processes are mathematically modeled by a highly nonlinear system of coupled partial differential equations. The nonlinearity of the system makes the design and implementation of robust numerical solvers a challenging task. In this work we consider the flow of two immiscible and incompressible fluids within a non‐deformable porous medium. A mixed pressure‐saturation formulation is adopted, allowing the transition from the unsaturated to saturated zones and maintaining numerical mass conservation. A cell‐centered finite volume method and an implicit Euler scheme are considered for the spatial and time discretization of the problem. In this work, we propose a solution method for two‐phase flow problems which is based on the combination of the modified Picard linearization method and a very simple cell‐centered multigrid algorithm that performs efficiently even for heterogeneous random media. This is shown in the numerical experiments, where two test problems are presented to demonstrate the robustness of the proposed solver.

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On Local Fourier Analysis of Multigrid Methods for PDEs with Jumping and Random Coefficients

Cited by 21 publications

References 53 publications

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems

A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

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