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

Abstract

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.

Multi-fidelity microstructure-induced uncertainty quantification by advanced Monte Carlo methods

“…Besides MLMC and MIMC, many other methods are also available in the literature; examples include, but are not limited to, multi-fidelity Monte Carlo (MFMC) [96,97,98], approximate control variate generalization of MFMC [99], multigrid (quasi-) Monte Carlo [100,101], multi-index stochastic collocation [102,103,104]. It should be noted that machine learning predictions can play a role of low-fidelity with low computational cost and relatively high error, as demonstrated in one of our previous studies [105].…”

GaussianRandomFields.jl: A Julia package to generate and sample from Gaussian random fields

“…GaussianRandomFields.jl provides Julia implementations of Gaussian random fields with stationary separable and non-separable isotropic and anisotropic covariance functions. It has been used in a number of recent works, including (Blondeel et al, 2020), (Robbe et al, 2021) and (Wu et al, 2023).…”

Multigrid methods

“…The special issue for this years conference consists of the nine papers 1‐9 . In Reference 1, Murray and Weinzierl develop a stabilized asynchronous FAC Multigrid solver for spacetrees, that is, meshes as they are constructed from octrees and quadtrees.…”

“…Multilevel algorithms for computing graph embeddings are developed by Quiring and Vassilevski in the paper 3 . In Reference 4, Robbe et al present an unbiased multiindex Monte Carlo method that reuses the Multiple Semi‐Coarsened Multigrid method to find coarse‐scale solutions to resolve the high anisotropy that can arise in random fields, e.g., permeability fields. In the paper, 5 Lee introduces a new algebraic multigrid method for solving systems of elliptic boundary‐value problems.…”