High Resolution Finite Difference and Spectral Algorithms for Piecewise Smooth Data

我们计划开发和实施新的高分辨率有限差分和频谱算法，减少伪吉布斯效应附近的内部边缘的分段规则的数据，并恢复高分辨率的底层信息之间的这些边缘。这些正是挑战经典方法的问题，并且在各种应用中具有很大的研究兴趣。教授E。Tadmor（UCLA）和A.盖尔布（亚利桑那州立大学），将继续他们正在进行的合作研究如下。(i)使用边缘检测和高分辨率重建技术在一维和多维空间中精确实现分段平滑数据。在这种情况下，我们将开发，分析和实施一个光谱准确的恢复程序相结合的本地化的基础上适当的浓度内核，确定numerous边缘，其次是一个新的双参数家庭的光谱软化恢复边缘之间的数据与指数精度。这些技术是现代高分辨率算法的核心。(ii)在过去的十年中，中心格式被证明是一个非常强大的，通用的工具，解决一般的非线性对流扩散问题。我们将整合新的恢复程序，并引入进一步的非笛卡尔增强程序的多维中心计划的决议。(iii)稳定和无杂散的光谱粘度算法的进一步发展和应用。特别是，我们计划将新的增强SV程序应用于由于混合和不稳定性而形成分段平滑的问题（Richtmyer-Meshkov，Taylor，...），浅水方程的数值模拟，混合型Euler-Poisson方程临界阈值现象的研究。看看你的周围：边缘无处不在。我们遇到的大部分数据--从图像到信号--都是分段平滑的，也就是说，它由被尖锐的内部边缘分隔开的平滑片段组成。该项目建议继续开发新的方法，以解决与包含分段平滑数据的问题相关的典型困难。具体来说，我们建议扩展我们正在进行的研究，处理重建的分段光滑数据的光谱信息。在这里，由平滑的“碎片”表示的大尺度通过各种非振荡重建来解决。由于未解决的小尺度，这是本地化的边缘附近的寄生振荡的困难。这个问题经常被认为是所谓的“振铃现象”。为此，检测边缘的位置，并在“平滑方向”上处理信息，即，远离检测到的边缘。应用包括高分辨率恢复在磁共振成像（MRI）、正电子发射断层扫描（PET）、气候学数据等逆应用中获得的分段平滑数据。分段光滑的数据也出现在各种时间相关的问题，由于自发破碎波图案。在这种情况下，困难变得更加复杂，因为必须跟踪移动边缘。我们还计划实施新的恢复程序来跟踪波的崩溃的演变，并将这些恢复程序与中心计划和光谱粘度方法-我们以前开发的两种现代高分辨率方法相结合。

We plan to develop and implement new high resolution finite difference and spectral algorithms which reduce the spurious Gibbs effects near the internal edges of piecewise regular data, and recover with high resolution the underlying information in between those edges. These are precisely the issues which defy classical methods and are of great research interest in various applications. Professors E. Tadmor (UCLA) and A. Gelb (ASU), will continue their ongoing cooperative research on the following. (i) Accurate realization of piecewise smooth data in one- and several space dimensions using edge detection and high-resolution reconstruction techniques. In this context we will develop, analyze and implement a spectrally accurate recovery procedures by combining localization based on appropriate concentration kernels which identify finitely many edges, followed by a novel two-parameter family of spectral mollifiers which recover the data between the edges with exponential accuracy. These techniques are at the heart of the modern high-resolution algorithms described below. (ii) Over the last decade, central schemes proved to be an extremely robust, all-purpose tool for solving general nonlinear convective-diffusive problems. We will integrate new recovery procedures and introduce further non-Cartesian enhancement procedures to the resolution of multidimensional central schemes. (iii) Further developments and applications of stable and spurious-free spectral viscosity algorithms. In particular, we plan to apply the new enhanced SV procedure to problems where piecewise smoothness forms due to mixing and instability (Richtmyer-Meshkov, Taylor, ...), simulations of the shallow water equations, and study of the critical threshold phenomena in mixed-type Euler-Poisson equations. Look around you: edges are everywhere. Much of the data we encounter -- from images to signals is piecewise smooth, that is, it consists of smooth pieces separated by sharp internal edges. This project proposes to continue the development of novel methods that combat the typical difficulties associated with problems containing piecewise smooth data. Specifically, we propose to extend our ongoing study of dealing with the reconstruction of piecewise smooth data from its spectral information. Here, the large scales represented by the smooth 'pieces' are resolved by a variety of non-oscillatory reconstructions. The difficulty arises with the spurious oscillations due to unresolved small scales which are localized in the neighborhood of the edges. The problem is often realized in terms of the so called 'ringing phenomena'. To this end, the location of the edges is detected and information is treated in the 'direction of smoothness', i.e., away from the detected edges. Application include the high resolution recovery of piecewise smooth data obtained in such inverse applications as magnetic resonance imaging (MRI), positron emission tomography (PET), climatology data and more. Piecewise smooth data also arise in various time dependent problems, due to spontaneous breakup in waves patterns. In this case, the difficulties become even more intricate, as one has to trace moving edges. We also plan to implement the new recovery procedures for tracing the evolution of breakdown of waves, and integrate these recovery procedures with central schemes and spectral viscosity methods -- two modern high resolution methods developed by us earlier.