波动方程的Curvelet表示是最优稀疏的
The Curvelet Representation of Wave Propagators is Optimally Sparse
摘要：
本文认为，曲波提供了一个强大的工具，表示非常一般的线性对称系统的双曲型微分方程。曲波是最近发展起来的一种多尺度系统，其中的单元在细尺度上是高度各向异性的，在细尺度上具有根据抛物尺度原理宽度~长度^[2]形成的有效支撑。我们证明了，对于广泛的一类线性双曲型微分方程，曲波表示的解决方案运营商是最佳稀疏和良好的组织。它是稀疏的，因为矩阵元素几乎以指数速度衰减（即比任何负多项式都快）;它是有组织的，因为极少数不可忽略的元素出现在移动的对角线附近。事实上，我们实际上表明，波群对曲线波的作用可以通过简单地沿着汉密尔顿流平移曲线波的中心来很好地近似--因此曲线波表示中的对角位移。这个结果的物理解释是，曲波可以被视为具有足够频率局部化的相干波形，使得它们表现得像波，但同时具有足够的空间局部化，使得它们同时表现得像粒子。
英文摘要：
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 in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ~ length^2 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 shifted diagonal. Indeed, we actually show that the action of the wave-group on a curvelet is well-approximated by simply translating the center of the curvelet along the Hamiltonian flow--hence the diagonal shift in the curvelet epresentation. 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.
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