  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.
