A point-free theory of quantitative homogenization
Abstract
We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and a flux corrector obtained by inverting the microscopic derivative on the residual field; this controls the error for regular macroscopic data in terms of explicit, ε-independent operator norms. This abstract perspective places on one footing stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media---and their respective convergence rates---are captured by the behavior of the spectral measure of the microscopic derivative operator, evaluated on the residual field, near zero frequency; in the periodic case the rate O(ε) is proved, while in the stochastic case the rates are conditional on a density-of-states hypothesis. Furthermore, our results apply for multiscale systems that do not uphold subadditivity of energy quantities or do not have a well-defined Brillouin zone.
Related
- Quantitative periodic homogenization and uniform regularity for nonlocal equations
- Quantitative homogenization of Monge-Ampère equations in periodic media
- Quantitative homogenization and large-scale regularity for nondivergence-form equations under a critical ellipticity moment
- Quantitative Homogenization and Uniform Lipschitz Estimates for Lamé-Stokes Systems
- Quantitative averaging of Markov-modulated additive functionals and Wentzell boundary homogenization
BibTeX
@misc{dang2026pointfree,
title = {{A point-free theory of quantitative homogenization}},
author = {Thuyen Dang and Yuliya Gorb and Silvia Jiménez Bolaños},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.05077},
eprint = {2608.05077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05077},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A point-free theory of quantitative homogenization
AU - Thuyen Dang
AU - Yuliya Gorb
AU - Silvia Jiménez Bolaños
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.05077
UR - https://arxiv.org/abs/2608.05077
AB - We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and a flux corrector obtained by inverting the microscopic derivative on the residual field; this controls the error for regular macroscopic data in terms of explicit, ε-independent operator norms. This abstract perspective places on one footing stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media---and their respective convergence rates---are captured by the behavior of the spectral measure of the microscopic derivative operator, evaluated on the residual field, near zero frequency; in the periodic case the rate O(ε) is proved, while in the stochastic case the rates are conditional on a density-of-states hypothesis. Furthermore, our results apply for multiscale systems that do not uphold subadditivity of energy quantities or do not have a well-defined Brillouin zone.
ER -