Numerical homogenization of spatial network models

TL;DR

This paper introduces a numerical homogenization method for spatial network models, enabling efficient coarse-scale simulations of processes like diffusion and deformation with proven convergence.

Contribution

It develops a novel approach using local subgraph problems to construct low-dimensional solution spaces for accurate, low-cost network modeling.

Findings

Method achieves optimal convergence under mild assumptions.

Numerical experiments confirm effectiveness for heat conduction and structural models.

Enables efficient simulations of complex network processes.

Abstract

We present and analyze a methodology for numerical homogenization of spatial networks, modelling e.g. diffusion processes and deformation of mechanical structures. The aim is to construct an accurate coarse model of the network. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data at a low computational cost. We prove optimal convergence of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (structural) models.