TY - JOUR

T1 - Spatio-spectral concentration of convolutions

AU - Hanasoge, Shravan M.

N1 - Funding Information:
The author acknowledges support from consultancy project PT54324 with Shell India , Ramanujan fellowship SB/S2/RJN-73/2013, the Max-Planck Partner Group Program and the NYU Abu Dhabi Center for Space Science . The work has benefited substantially from numerous conversations, and SMH gratefully acknowledges discussions with Alain Plattner, Srinivasa Varadhan, Rishi Sharma and Sandip Trivedi.
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/5/15

Y1 - 2016/5/15

N2 - 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.

AB - 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.

KW - Homogenisation

KW - Renormalisation group

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U2 - 10.1016/j.jcp.2016.02.068

DO - 10.1016/j.jcp.2016.02.068

M3 - Article

AN - SCOPUS:84960156718

VL - 313

SP - 674

EP - 686

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -