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Sparse Upscaling Operators for Homogenization*

by V.H. Hoang and C.H. Pang and Ch. Schwab

(Report number 2026-15)

Abstract
For linear, elliptic divergence-form PDE with \(2\)-scale coefficients \({\boldsymbol A}(x,y)\) taking values \(x\) in a bounded polytopal domain \({\mathrm D}\subset {\mathbb R}^d\) and \(y\) in unit cell \(Y = (0,1)^d\), \(d\geq 1\), \(Y\)-periodic with respect to \(y\), we propose and analyze a novel, sparse-tensor algorithm for the efficient computation of (a numerical approximation of) the homogenized or upscaled coefficient matrix \({\boldsymbol A}^0(x)\) throughout the domain \({\mathrm D}\). We numerically approximate the homogenization map \({\boldsymbol H}: W^{s,p}({\mathrm D} ;W^{t,\infty}_{per}(Y)^{d\times d}) \to W^{s,p}({\mathrm D};{\mathbb R}^{d\times d}): {\boldsymbol A}(x,y) \mapsto {\boldsymbol A}^0(x) \) for \(s,t \in \{0,1,2\}\), \(p\in [1,\infty]\) from the \(2\)-scale coefficient \({\boldsymbol A}(x,y) \in {W}^{{s,p}}({\mathrm D} ; W^{t,\infty}_{per}(Y)^{d\times d})\) to the upscaled (homogenized) macroscopic diffusion coefficient \({\boldsymbol A}^0\in W^{s,p}({\mathrm D};{\mathbb R}^{d\times d})\). A first result is that \({\boldsymbol H}\) is analytic as a map between these spaces. This implies existence of sparse polynomial and neural network approximations of \({\boldsymbol H}\), via parametric encodings of \({\boldsymbol A}\) in \({\mathrm D}\times Y\). A second result pertains to fully discrete approximations \({\boldsymbol H}_L\) of \({\boldsymbol H}\). Here, \({\boldsymbol A}\) is efficiently evaluated on sparse sets of query points in \(D\times Y\). These sets correspond to vertices resp. barycenters of simplices \(T\) in a regular, quasi-uniform triangulation \({\mathcal T}^L\) of \({\mathrm D}\subset {\mathbb R}^d\) into \(N_L = \#({\mathcal T}^L) = O(h_L^{-d})\) many simplices \(T\in {\mathcal T}^L\) with meshwidth \(h_L = O(2^{-L}) >0\). Under provision of suitable regularity, the proposed algorithm accesses the \(2\)-scale coefficient \({\boldsymbol A}\) in \(O(N_L \log N_L)\) many points of a sparse grid in \({\mathrm D} \times Y\) whilst providing an approximation of the homogenized coefficient \({\boldsymbol A}^0\) in \({\mathrm D}\) with accuracy \(O(h_L^{s})\) in \(L^\infty({\mathrm D})\) and overall work scaling as \(O(N_L\log N_L) = O(h_L^{-d}|\log(h_L)|)\). %A similar result holds for \(\mathbf{N}(x,y)\) %in the first order corrector function %\(u^1(x,y) = \mathbf{N}(x,y) \cdot \nabla_x u^0(x)\). The approximation in \(L^\infty({\mathrm D})\) is \(O(h_L^s) = O(N_L^{-s/d})\) accurate, generally, where \(0< s \leq 2\) is determined by the \(2\)-scale coefficient, and limited by the order of the \({\mathbb P}_1\)-FEM employed for solving the cell problem. %\(2\)-scale FEM. Two architectures of neural upscaling operator surrogates are proposed: an encoder-approximator-decoder architecture, and a U-Net type architecture. Both access the \(2\)-scale coefficient \({\boldsymbol A}\) in \(O(N_L \log N_L)\) query points, with log-linear error vs. network-size bounds. Numerical experiments confirm the theoretical error estimates of the sparse tensor upscaling. The present error analysis extends to higher order under provision of (a) stronger regularity of \({\boldsymbol A}\) and (b) higher order, Lagrangian FEM.

Keywords:

@Techreport{HPS26_1170,
  author = {V.H. Hoang and C.H. Pang and Ch. Schwab},
  title = {Sparse Upscaling Operators for Homogenization*},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-15.pdf },
  year = {2026}
}

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