"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.

自从发现小波以来，过去几十年来看到了有关功能或分布的新表示系统设计的爆炸。这项工作的主要目的是找到表示各种信号类别稀疏近似的表示系统，其主要应用位于信号处理区域。作为一个例子，我们提到了与标准离散化方法（如有限的元素或小波）相比，剪切，孔子或ridgelet系统可以执行的定向信息的有效检测。考虑到这些新表示系统的近似特性的各种壮观的结果，下一步也是一个自然的下一步是将它们用于操作员方程的数值处理。但是，基于这些新表示系统的数值方法的开发当前是面对瓶颈的，迄今不存在这些代表系统的有用构造。该项目的目的是通过在有限域上构建和分析新的离散表示系统来删除此瓶颈，从而享受与剪切相同的最佳近似属性，同时仍构成了大型PDE的能量空间的稳定离散化（例如，Sobolev Spaces）。我们的重点将放在综合理论上，以适应有限领域的功能空间。我们将特别感兴趣这些空间的离散表征，这些空间上各种PDE的规律性理论以及各种PDE的Galerkin矩阵的可压缩性。预计这些研究将为操作员方程的随后开发和实施一系列新型的离散方法奠定基础，这些方程的表现优于当前小波或有限元的基于有限元的方法，用于大量重要问题，例如反应扩散方程，线性传输方程或具有不连续扩散系数的椭圆形PDE。