# Enhanced Multi-Index Monte Carlo by means of Multiple Semi-Coarsened Multigrid for Anisotropic Diffusion Problems

## TL;DR

This paper introduces an enhanced Multi-Index Monte Carlo method that leverages Multiple Semi-Coarsened Multigrid techniques for efficient and unbiased uncertainty quantification in anisotropic diffusion problems, demonstrating improved performance and robustness.

## Contribution

It develops a novel unbiased MIMC estimator that reuses MSG coarse solutions, unifying adaptive MIMC and unbiased estimation, with theoretical cost analysis and numerical validation.

## Key findings

- The new estimator is more robust than previous methods.
- Numerical experiments show improved efficiency in anisotropic random fields.
- The method effectively handles hyperparameters in covariance models.

## 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.