Multiscale Graph Reduction for Heterogeneous and Anisotropic Discrete Diffusion Processes

We present multiscale graph-based reduction algorithms for upscaling heterogeneous and anisotropic diffusion problems. The proposed coarsening approaches begin by constructing a partitioning of the computational domain into a set of balanced local subdomains, resulting in a standard type of domain decomposition. Given this initial decomposition, general coarsening techniques based on spectral clustering are applied within each subgraph in order to accurately identify the key microscopic features of a given system. The spectral clustering algorithm is based on local generalized eigen-decompositions applied to the signed graph Laplacian. The resulting coarse-fine splittings are combined with two variants of energy-minimizing strategies for constructing coarse bases for diffusion problems. The first is an unconstrained minimization formulation in which local harmonic extensions are applied column-wise to construct multi-vector preserving interpolation in each region, whereas the second approach is a variant of the constrained energy minimization formulations derived in the context of non-local multi-continua upscaling techniques. We apply the resulting upscaling algorithms to a variety of tests coming from the graph Laplacian, including diffusion in the perforated domain, channelized media, highly anisotropic settings, and discrete pore network models to demonstrate the potential and robustness of the proposed coarsening approaches. We show numerically and theoretically that the proposed approaches lead to accurate coarse-scale models.