  In this work, we propose a generalized multiscale inversion algorithm for
heterogeneous problems that aims at solving an inverse problem on a
computational coarse grid. Previous inversion techniques for multiscale
problems seek a coarse-grid media properties, e.g., permeability and
conductivity, and by doing so, they assume that there exists a homogenized
representation of the underlying fine-scale permeability field on a coarse
grid. Generally such assumptions do not hold for highly heterogeneous fields,
e.g., fracture media or channelized fields, where the width of channels are
very small compared to the coarse-grid sizes. In these cases, grid refinement
can lead to many degrees of freedom, and thus unattractive to apply. The
proposed algorithm is based on the Generalized Multiscale Finite Element Method
(GMsFEM), which uses local spectral problems to identify non-localized
features, i.e., channels (high-conductivity inclusions that connect the
boundaries of the coarse-grid block). The inclusion of these features in the
coarse space enables one to achieve a good accuracy. The approach is valid
under the assumption that the solution can be well represented in a
reduced-dimensional space by multiscale basis functions. In practice, these
basis functions are non-observable as we do not identify the fine-scale
features of the permeability field. Our inversion algorithm finds the
discretization parameters of the resulting system. By doing so, we identify the
appropriate coarse-grid parameters representing the permeability field instead
of fine-grid permeability field. We illustrate the approach by numerical
results for fractured media.
