High Resolution Finite Difference and Spectral Algorithms for Piecewise Smooth Data

分段平滑数据的高分辨率有限差分和谱算法

• 批准号: 0107428
• 负责人: Eitan Tadmor
• 金额: $ 20.62万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107428&HistoricalAwards=false
• 关键词: Resolution; Finite; Difference; Spectral; Algorithms

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.

我们计划开发和实施新的高分辨率有限差分和谱算法，以减少分段规则数据内部边缘附近的寄生吉布斯效应，并以高分辨率恢复这些边缘之间的基础信息。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。 E. Tadmor 教授（加州大学洛杉矶分校）和 A. Gelb 教授（亚利桑那州立大学）将继续开展以下方面的合作研究。 （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准确实现分段平滑数据。在这种情况下，我们将开发、分析和实现光谱精确的恢复程序，方法是结合基于适当浓度核的定位，识别有限多个边缘，然后是一个新颖的双参数光谱缓和器系列，以指数精度恢复边缘之间的数据。这些技术是下述现代高分辨率算法的核心。 (ii) 在过去的十年中，中心方案被证明是解决一般非线性对流扩散问题的极其强大的通用工具。我们将集成新的恢复程序，并引入进一步的非笛卡尔增强程序来解决多维中心方案。 (iii) 稳定且无杂散的光谱粘度算法的进一步开发和应用。特别是，我们计划将新的增强型 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 等）、浅水方程的模拟以及混合型 Euler-Poisson 方程中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，它由由尖锐的内部边缘分隔的平滑片段组成。该项目建议继续开发新方法，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处理从光谱信息重建分段平滑数据的研究。在这里，由平滑“碎片”代表的大尺度通过各种非振荡重建来解决。由于位于边缘附近的未解决的小尺度而导致寄生振荡，因此出现了困难。该问题通常通过所谓的“振铃现象”来实现。为此，检测边缘的位置，并在“平滑方向”（即远离检测到的边缘）处理信息。应用包括对磁共振成像 (MRI)、正电子发射断层扫描 (PET)、气候数据等逆向应用中获得的分段平滑数据进行高分辨率恢复。由于波浪模式的自发破裂，分段平滑数据也会出现在各种与时间相关的问题中。在这种情况下，困难变得更加复杂，因为人们必须追踪移动的边缘。我们还计划实施新的恢复程序来追踪波浪分解的演化，并将这些恢复程序与中心方案和光谱粘度方法（我们之前开发的两种现代高分辨率方法）相结合。