Generalized Multiscale Inversion for Heterogeneous Problems (Communications in Computational Physics, 2018)
In this work, we propose a generalized multiscale inversion algorithm forheterogeneous problems that aims at solving an inverse problem on a computationalcoarse grid. Previous inversion techniques for multiscale problems seek a coarse-gridmedium properties, e.g., permeability and conductivity, and by doing so, they assumethat 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 channelsare very small compared to the scale of coarse grids. In these cases, grid refinementcan lead to many degrees of freedom, and thus numerically unattractive to apply. Theproposed 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-gridblock). The inclusion of these features in the coarse space enables one to achieve agood accuracy. The approach is valid under the assumption that the solution can bewell represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify thefine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach bynumerical results for fractured media.
Keywords: Multiscale inversion; multiscale problem; generalized multiscale finite element method; coarse-grid.