Nonuniform sampling and multiscale computation

Abstract

In homogenization theory and multiscale modeling, typical functions satisfy the scaling law $f^ε(x) = f(x,x/ε)$, where $f$ is periodic in the second variable and $ε$ is the smallest relevant wavelength, $0<ε\ll1$. Our main result is a new $L^{2}$-stability estimate for the reconstruction of such bandlimited multiscale functions $f^ε$ from periodic nonuniform samples. The goal of this paper is to demonstrate the close relation between and sampling strategies developed in information theory and computational grids in multiscale modeling. This connection is of much interest because numerical simulations often involve discretizations by means of sampling, and meshes are routinely designed using tools from information theory. The proposed sampling sets are of optimal rate according to the minimal sampling requirements of Landau \cite{Landau}.