Spatio-spectral concentration of convolutions

Research output: Contribution to journalArticle

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 homogenisation 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 Renormalisation 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.

Original languageEnglish (US)
Pages (from-to)674-686
Number of pages13
JournalJournal of Computational Physics
Volume313
DOIs
StatePublished - May 15 2016

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Convolution
convolution integrals
Mathematical operators
coefficients
cut-off
scale models
Cutoff frequency
Linear equations
Differential equations
operators
linear equations
tradeoffs
homogenizing
elimination
differential equations
grids
filters

Keywords

  • Homogenisation
  • Renormalisation group

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy (miscellaneous)

Cite this

Spatio-spectral concentration of convolutions. / Hanasoge, Shravan.

In: Journal of Computational Physics, Vol. 313, 15.05.2016, p. 674-686.

Research output: Contribution to journalArticle

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