Nonuniform sampling and multiscale computation

TL;DR

This paper establishes a new $L^{2}$-stability estimate for reconstructing multiscale functions from nonuniform samples, linking sampling theory with multiscale modeling and demonstrating optimal sampling rates.

Contribution

It introduces a novel stability estimate for multiscale function reconstruction from nonuniform samples, bridging information theory and computational grids.

Findings

Proves a new $L^{2}$-stability estimate for multiscale functions.

Shows the sampling sets are of optimal rate according to Landau.

Establishes a connection between sampling strategies and multiscale modeling.

Abstract

In homogenization theory and multiscale modeling, typical functions satisfy the scaling law $f^{\epsilon}(x) = f(x,x/\epsilon)$, where $f$ is periodic in the second variable and $\epsilon$ is the smallest relevant wavelength, $0<\epsilon\ll1$. Our main result is a new $L^{2}$-stability estimate for the reconstruction of such bandlimited multiscale functions $f^{\epsilon}$ 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}.