"New Smoothness Spaces on Domains and Their Discrete Characterization"

“域上的新平滑空间及其离散特征”

• 批准号: 373295677
• 负责人: Professor Dr. Stephan Dahlke
• 金额: --
• 依托单位: Arbeitsgruppe Numerik (Dahlke)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/373295677?language=en
• 关键词: New; Smoothness; Spaces; Domains; Their

Since the discovery of wavelets, the past decades have seen an explosion concerning the design of novel representation systems for functions or distributions. The main intent of that work has been to find representation systems which are optimal for the sparse approximation of various signal classes with the main applications lying in the area of signal processing. As an example we mention the efficient detection of directional information that can be performed by shearlet or curvelet or ridgelet systems which possess vastly superior approximation properties as compared to standard discretization methods such as finite elements or wavelets. Having the various spectacular results concerning the approximation properties of these new representation systems in mind, a natural next step would be to employ them also for the numerical treatment of operator equations.However, the development of numerical methods based on these new representation system is currently facing the bottleneck that to date no useful constructions of these representation systems on bounded domains exist. The aim of this project is to remove this bottleneck by constructing and analyzing new discrete representation systems on finite domains which on the one hand enjoy the same optimal approximation properties as, for example, shearlets while still forming a stable discretization of the energy space for a large class of PDEs (for example Sobolev spaces). Our focus will be on the development of a comprehensive theory for the adaption of function spaces to finite domains. We will be especially interested in the discrete characterization of such spaces, the regularity theory of various PDEs on such spaces and the compressibility properties of Galerkin matrices of various PDEs with respect to these discretizations. These studies are expected to lay the ground work for the subsequent development and implementation of a large class of novel discretization methods for operator equations which outperform current wavelet- or finite-element-based methods for a large class of important problems such as reaction-diffusion equations, linear transport equations or elliptic PDEs with discontinuous diffusion coefficients.

自从小波被发现以来，在过去的几十年里，关于函数或分布的新表示系统的设计出现了爆炸式的增长。这项工作的主要目的是找到最佳的表示系统的稀疏近似的各种信号类的主要应用在于在信号处理领域。作为一个例子，我们提到的方向信息的有效检测，可以由剪切波或曲波或脊波系统，拥有大大优于上级近似性能相比，标准的离散化方法，如有限元或小波。有各种壮观的结果，这些新的表示系统的近似性能的想法，一个自然的下一步将是采用它们也数值处理的算子方程，然而，基于这些新的表示系统的数值方法的发展目前面临的瓶颈，到目前为止，没有有用的结构，这些表示系统的有界域上存在。该项目的目的是通过构建和分析有限域上的新的离散表示系统来消除这一瓶颈，该系统一方面具有与剪切波相同的最佳逼近特性，同时仍然形成了一大类偏微分方程（例如Sobolev空间）的能量空间的稳定离散化。我们的重点将是发展一个全面的理论，为适应功能空间有限域。我们将特别感兴趣的离散表征这样的空间，正则性理论的各种偏微分方程在这样的空间和可压缩性的Galerkin矩阵的各种偏微分方程相对于这些离散。这些研究预计将奠定基础工作，为后续开发和实施的一大类新的离散化方法的算子方程，优于目前的小波或有限元为基础的方法，一大类重要的问题，如反应扩散方程，线性输运方程或椭圆偏微分方程的不连续扩散系数。