Spatio-spectral concentration of convolutions

TL;DR

This paper develops a spectral homogenization method using Fourier basis to create coarse-scale models of coefficients in differential equations, balancing accuracy and spectral cutoff.

Contribution

It introduces a Fourier-based numerical homogenization approach for deterministic linear equations to model sub-grid-scale coefficients at specific frequency cutoffs.

Findings

The method effectively reproduces coarse solution scales in 1D and 2D.

Accuracy depends on the spectral cutoff and basis choice.

The approach relates to mode-elimination RG and highlights a tradeoff governed by an uncertainty principle.

Abstract

Differential equations may possess coefficients that vary on a spectrum of scales. Because coefficients are typically multiplicative in real space, they turn into convolution operators in spectral space, mixing all wavenumbers. However, in many applications, only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse-scale models of the coefficients in such a manner that the large scales of the solution are left intact. Here we apply the method of numerical homogenization to deterministic linear equations to generate sub-grid-scale models of coefficients at desired frequency cutoffs. We use the Fourier basis to project, filter and compute correctors for the coefficients. The method is tested in 1D and 2D scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff. We relate this method to mode-elimination Renormalization Group (RG) and discuss the connection between accuracy and the cutoff wavenumber. The tradeoff is governed by a form of the uncertainty principle for convolutions, which states that as the convolution operator is squeezed in the spectral domain, it broadens in real space. As a consequence, basis sparsity is a high virtue and the choice of the basis can be critical.