Candès, Emmanuel J. and Demanet, Laurent (2005) The curvelet representation of wave propagators is optimally sparse. Communications on Pure and Applied Mathematics, 58 (11). pp. 14721528. ISSN 00103640. http://resolver.caltech.edu/CaltechAUTHORS:CANcpam05

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Abstract
This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [7, 9] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width length2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. * It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial) * and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows  hence the diagonal shifts in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
Item Type:  Article 

Additional Information:  This research was supported by a National Science Foundation grant DMS 0140698 (FRG), by a DOE grant DEFG0302ER25529, and by an Alfred P. Sloan Fellowship. E. C. would like to thank Guillaume Bal for valuable conversations. Author preprint 
Subject Keywords:  Hyperbolic Equations, Waves, Hamiltonian Equations, Characteristics, Geometrical Optics, Fourier Integral Operators, Curvelets, Sparsity, Nonlinear Approximation, Multiscale Representations, Parabolic Scaling 
Record Number:  CaltechAUTHORS:CANcpam05 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:CANcpam05 
Alternative URL:  http://dx.doi.org/10.1002/cpa.20078 
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  2038 
Collection:  CaltechAUTHORS 
Deposited By:  Archive Administrator 
Deposited On:  04 Mar 2006 
Last Modified:  26 Dec 2012 08:47 
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