Part IX: Multi-Scale Methods
Bridging micro and macro — homogenization proving CA\(\to\)PDE convergence with \(D = \text{spread}/4\), renormalization group deriving \(\beta=3/4\) from RG fixed points, and equation-free methods for coarse projective integration.
Part Overview
The central challenge of urban modeling: how do microscopic rules produce macroscopic laws? This part provides three rigorous answers. Homogenization extracts effective PDEs from cellular automata, renormalization group methods identify universal scaling exponents at fixed points, and equation-free methods simulate macro-dynamics without ever writing down the macro-equation.
Key Topics
- • Homogenization CA\(\to\)PDE
- • Two-scale expansion
- • Cell problem
- • Effective diffusivity tensor
- • RG flow and fixed points
- • Kevrekidis lift-evolve-restrict
- • Fisher-Rao geometry
3 chapters | Scale bridging | From lattice rules to continuum laws
Chapters
Chapter 1: Homogenization (CA\(\to\)PDE)
Two-scale asymptotic expansion proving that cellular automaton dynamics converge to a diffusion PDE. Solving the cell problem yields the effective diffusivity \(D = \text{spread}/4\), rigorously connecting discrete and continuum descriptions.
Chapter 2: Renormalization Group
Coarse-graining urban models under scale transformations. RG flow equations, fixed points, and the derivation of the universal scaling exponent \(\beta = 3/4\) governing urban growth near criticality.
Chapter 3: Equation-Free Methods
Kevrekidis lift-evolve-restrict framework for coarse projective integration. Simulate the microscopic model briefly, extract macroscopic observables, and project forward in time — without ever writing the macro-equation. Fisher-Rao geometry guides the coarse variable selection.