Emmanuel J. Candes and Laurent Demanet
In this tutorial lecture we review the recent result that curvelets provide a powerful
tool for representing very general linear symmetric systems of hyperbolic differential
equations. Curvelets are a recently developed multiscale system [8, 5] 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. For a wide class
of hyperbolic equations, including the usual wave equation arising in acoustics and
electromagnetism, 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 shifted diagonal.
Indeed, the action of the wave-group on a curvelet is well-approximated by simply
translating the center of the curvelet along the Hamiltonian °ow|hence the diagonal
shift 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.
These notes are a shortened non-technical version of the reference paper [3], which
can be downloaded at
http://www.acm.caltech.edu/~demanet/pubs.htm
Keywords. Hyperbolic Equations, Waves, Hamiltonian Equations, Characteristics,
Geometrical Optics, Fourier Integral Operators, Curvelets, Sparsity, Nonlinear Approxima-
tion, Multiscale Representations, Parabolic Scaling.
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