Multilevel Monte Carlo Approaches for Numerical Homogenization

International audienceIn this article, we study the application of Multi-Level Monte Carlo (MLMC) approaches to numerical random homogenization. Our objective is to compute the expectation of some functionals of the homogenized coefficients, or of the homogenized solutions. This is accomplished within MLMC by considering different levels of representative volumes (RVE), and, when it comes to homogenized solutions, different levels of coarse-grid meshes. Many inexpensive computations with the smallest RVE size and the largest coarse mesh are combined with fewer expensive computations performed on larger RVEs and smaller coarse meshes. We show that, by carefully selecting the number of realizations at each level, we can achieve a speed-up in the computations in comparison to a standard Monte Carlo method. Numerical results are presented both for one-dimensional and two-dimensional test-cases

On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method

Mini-Workshop: Numerical Upscaling for Media with Deterministic and Stochastic Heterogeneity

This minisymposium was third in series of similar events, after two very successful meetings in 2005 and 2009. The aim was to provide a forum for an extensive discussion on the theoretical aspects and on the areas of application and validity of numerical upscaling approaches for heterogeneous problems with deterministic and stochastic coefficients. The intensive discussions during the meeting contributed to a better understanding of upscaling approaches for multiscale problems with stochastic coefficients, and for synergy between scientists coming to this topic from the area of deterministic multiscale problems on one hand, and those coming from the area of SPDE on the other hand. Recent advanced results on upscaling approaches for deterministic multiscale problems were presented, well mixed with strong presentations on SDE and SPDE. The open problems in these areas were discussed, with emphasis on the case of stochastic coefficients brainstorming numerous numerical upscaling approaches. A number of young researchers, very actively working in these areas, were involved in the workshop discussing the links between scales., thus ensuring the continuity between the generations of researchers

On the predictivity of pore-scale simulations : estimating uncertainties with multilevel Monte Carlo

A fast method with tunable accuracy is proposed to estimate errors and uncertainties in pore-scale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another “equivalent” sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an open-source code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics

An offline-online strategy for multiscale problems with random defects

In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good coarse-scale approximation of the solution for small $p$. Moreover, extensive numerical experiments illustrate that relative errors of a few percent are achieved up to at least $p = 0.1$. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation