自适应磨光器用于从谱信息高分辨率恢复分段光滑数据
Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information
AI总结：
本文提出自适应磨光器，通过边缘检测定位和高阶抵消，从谱信息高分辨率恢复分段光滑数据，实现指数精度并消除吉布斯振荡。
摘要：
我们讨论了从（伪）谱信息重建分段光滑数据的问题。当数据全局光滑时，谱投影具有优越的分辨率，而跳跃间断的存在会在边缘附近引起虚假的${\cal O}(1)$吉布斯振荡，并导致整体退化到不可接受的一阶收敛速度。我们的目标是在分段光滑情形下恢复这种优越的精度，这通过磨光来实现。\n\n这里我们使用了Gottlieb和Tadmor [GoTa85]引入的两参数族谱磨光器的改进版本。普遍使用的单参数有限阶磨光器基于膨胀。相比之下，我们的磨光器通过一个复杂的高阶抵消过程实现高分辨率。为此，我们首先使用边缘检测程序[GeTa00a, GeTa00b]实现定位步骤。然后，在远离边缘的光滑方向上执行分段光滑数据的精确恢复，自适应性负责实现高分辨率。由此产生的自适应磨光器大大加快了收敛速度，在指数精度内恢复分段解析数据，同时消除了[GoTa85]中残留的虚假振荡。因此，这些自适应磨光器为分段光滑数据的精确后处理提供了一种稳健、通用的“黑盒”程序。
英文摘要：
We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious ${\cal O}(1)$ Gibbs oscillations in the neighborhood of edges and an overall deterioration to the unacceptable first-order convergence rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation. In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancelation. To this end, we first implement a localization step using edge detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box'' procedure for accurate post processing of piecewise smooth data.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
