PDE on Graphs and Their Continuum Limits
summary
Shows when graph PDEs actually approximate a continuum PDE: with kernel weights and ε→0 the graph Laplacian equals a nonlocal integral operator and then converges to −ρ−1 div(ρ2∇·). Uses Bernstein concentration to control sampling noise, Taylor expansions for the nonlocal→local step, and a comparison principle plus barrier functions to get sup‑norm error O(ε) with high probability if ε ≳ N−1/(d+4). Variational problems use Gamma‑convergence to reach the harder threshold ε ≳ N−1/d.
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chapters (13)
- 0:00Introduction to PDE on Graphs
- 1:04Data Setup and Graph Construction
- 3:21Spectral Clustering Overview
- 7:57Graph Types and Scaling
- 11:02Graph Laplacian and PDE Models
- 15:13Graph Laplacian Definition
- 16:58Continuum Limits and Consistency
- 19:44Concentration Inequalities
- 25:15Consistency of the Graph Laplacian
- 31:21Convergence Analysis
- 39:01PDE Solution Estimates
- 43:37Viscosity Solutions and Consistency
- 49:01Final Proofs and Conclusions
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