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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.09280 |
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| _version_ | 1866917477814894592 |
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| author | Zhou, Yingjie Zhong, Xiang Ye, Changqing Chung, Eric T. |
| author_facet | Zhou, Yingjie Zhong, Xiang Ye, Changqing Chung, Eric T. |
| contents | 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 Poincaré 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models Zhou, Yingjie Zhong, Xiang Ye, Changqing Chung, Eric T. Numerical Analysis 65N12, 65N30, 05C82, 90C35 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 Poincaré 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. |
| title | Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models |
| topic | Numerical Analysis 65N12, 65N30, 05C82, 90C35 |
| url | https://arxiv.org/abs/2605.09280 |