Nonuniform Sampling and Multiscale Computation

  title={Nonuniform Sampling and Multiscale Computation},
  author={Bj{\"o}rn Engquist and Christina Frederick},
  journal={Multiscale Model. Simul.},
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… 

Figures from this paper

An L2-stability estimate for periodic nonuniform sampling in higher dimensions

On sequential multiscale inversion and data assimilation

A Complete Bibliography of Publications in Multiscale Modeling & Simulation

(BV, L) [TNV04]. 1 [BLO17, FG08, VS11]. 1 + 1 [PM14, SPM18, MT09]. 13 [Str05, Tor06]. 2 [AE11, DD13, FFJD09, JR03, VO13, YLY15, Yin15a]. 2 + 1 [BV06, MK06]. 3 [CLLW15, DWC15, LH14, PKC05]. 30

Multiscale Modeling & Simulation

Accuracy [Aar04]. Acoustic [VMK05]. Adaptive [JLT04]. Adjoint [CL03b]. Adsorbing [AMK03]. Algebraic [KCH03]. Algorithm [MS04]. Algorithms [BCM05]. Alloys [MR03]. Amplifiers [LM04]. Amplitude [AIL05].



Iterative reconstruction of multivariate band-limited functions from irregular sampling values

This paper describes a real analysis approach to the problem of complete reconstruction of a band-limited, multivariate function f from irregularly spaced sampling values $(f(x_i ))_{i \in I} $. The


A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set

Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces

A transformation method for the reconstruction of functions from nonuniformly spaced samples

A sampling theory which extends the uniform sampling theory of Whittaker et al. to include nonuniform sample distributions and shows that a more general result can be obtained by treating the sample sequence as the result of applying a coordinate transformation to the uniform sequence.

On Nonuniform Sampling of Bandwidth-Limited Signals

The purpose of this investigation is to examine four special nonuniform sampling processes in detail, and to deduce some interesting properties of bandwidth-limited signals. The main results are

Perfect reconstruction formulas and bounds on aliasing error in sub-nyquist nonuniform sampling of multiband signals

The problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples is examined and an explicit reconstruction formula is derived.

Nonuniform Sampling of Periodic Bandlimited Signals

Two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed and it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments.

Sampling-50 years after Shannon

  • M. Unser
  • Computer Science
    Proceedings of the IEEE
  • 2000
The standard sampling paradigm is extended for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets, and variations of sampling that can be understood from the same unifying perspective are reviewed.

Introduction to Fourier analysis and wavelets

This book provides a concrete introduction to a number of topics in harmonic analysis, accessible at the early graduate level or, in some cases, at an upper undergraduate level. Necessary

Sampling of Bandlimited Functions on Unions of Shifted Lattices

The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given, including a numerical example implemented in MATLAB.