Multiscale Graph Reduction for Heterogeneous and Anisotropic Discrete Diffusion Processes

TL;DR

This work tackles efficient upscaling of heterogeneous and anisotropic diffusion by introducing Multiscale Graph Reduction (MsGR), which blends balanced domain decomposition with local spectral clustering to identify multiple coarse-scale variables. Local communities are detected within subdomains via a generalized eigenproblem on the signed graph Laplacian, and coarse bases are constructed using two strategies: CF, which uses centroid-based coarse nodes and ideal interpolation, and MC, which treats clusters as coarse continua and solves constrained energy minimization. The coarse-scale system is assembled as $A_c = R A P$ and $f_c = R f$, with $u_{ms} = P u_c$, and theoretical error bounds link the coarse-space quality to subdomain size $H$ and intra-cluster contrast $C_{\text{ratio}}$, while numerical tests on perforated domains, highly anisotropic media, and pore networks demonstrate robust accuracy and efficiency. The results indicate that global MC often yields the best accuracy, while localization via oversampling and clustering can achieve substantial sparsification with careful choice of $M$ and $N_{\Omega}$, making MsGR a practical upscaling tool for complex discrete diffusion problems.

Abstract

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.

Figures (18)

• Figure 1: Example heterogeneous perforated domain $\Omega$ (right) and mesh $\mathcal{T}_h$ (left).
• Figure 2: Pore-scale structure with two scales and corresponding high-contrast network
• Figure 3: Graph partitioning and local domains $\Omega_k$ with $N_{\Omega}=100$ (left). Illustration of oversampled subdomain $\Omega_k^+$. Target subdomain $\Omega_k$ (middle) and oversampled subdomains $\Omega_k^+$ based on distances $\delta_H$ (right). Red color is used for target subdomain $\Omega_k$
• Figure 4: Illustration of spectral clustering and centroids for high-contrast case, $K(x)$ (blue: $K = 1$, red: $K = 10^4$). The first plot displays the coefficient field $K(x)$. The second, third, fourth and fifth plots show the resulting partitions for 4, 8, 16 and 32 clusters.
• Figure 5: Illustration of local clusters and their centroids for highly anisotropic case. Resulting partitions for 4, 8, 16 and 32 clusters (from left to right).

Theorems & Definitions (4)

• Lemma 1
• proof
• Theorem 1
• proof