Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models

TL;DR

This paper introduces an algebraic multiscale method for highly heterogeneous spatial networks that is efficient, geometry-independent, and adaptable to various boundary conditions, validated by theoretical analysis and numerical experiments.

Contribution

The proposed method is a purely algebraic multiscale approach that avoids geometric parameters, enabling broader applicability and efficient simulation of complex heterogeneous networks.

Findings

Convergence is independent of heterogeneity contrast.

The method operates entirely within an algebraic framework.

Numerical experiments confirm the method's effectiveness.

Abstract

Modeling complex spatial networks with multiscale heterogeneity poses significant mathematical and computational challenges. Lacking explicit PDE discretizations and facing excessive degrees of freedom, conventional methods often become computationally prohibitive. To address these challenges, we propose an efficient multiscale modeling for highly heterogeneous spatial networks. We construct multiscale basis functions tailored to spatial network models with heterogeneous edge weights and node degrees. A key novelty is that the proposed method doesn't introduce geometric parameters (such as Dirichlet nodes, distances, or mesh sizes), thereby preserving its purely algebraic nature and ensuring broad applicability. By incorporating a subgraph-wise estimate, we define a Poincar\'e constant $C_{\mathrm{po}}$ that renders the method independent of the underlying graph geometry. Then through an appropriate choice of the number of graph oversampling layers, we establish an $O(C_{\mathrm{po}})$ convergence independent of the local heterogeneity contrast. Notably, our scheme operates entirely within an algebraic framework, eliminating the need for Dirichlet nodes and positive-definiteness on specific matrices arising in the model. This flexibility enables the simulation of a wider range of physical models and accommodates various boundary conditions. Rigorous theoretical analyses are provided under suitable assumptions, and extensive numerical experiments validate the effectiveness of the proposed approach.