"New Smoothness Spaces on Domains and Their Discrete Characterization"

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型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适合各种信号类的稀疏逼近的表示系统，主要应用于信号处理领域。作为一个例子，我们提到了方向信息的有效检测，可以由shearlet或曲线或脊波系统执行，与有限元或小波等标准离散化方法相比，它们具有更优越的近似特性。有了关于这些新的表示系统的近似性质的各种惊人的结果，下一步自然是将它们也用于算子方程的数值处理。然而，基于这些新的表示系统的数值方法的发展目前面临着瓶颈，即迄今为止还没有有效的有界域上的这些表示系统的结构。本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消除这一瓶颈，该系统一方面具有与shearlet相同的最佳近似性质，同时仍然形成一个稳定的能量空间离散化，用于大类别的偏微分方程（例如Sobolev空间）。我们的重点将是发展一个全面的理论，以适应函数空间的有限域。我们将特别感兴趣的是这些空间的离散表征，这些空间上各种偏微分方程的正则性理论，以及各种偏微分方程的伽辽金矩阵关于这些离散化的可压缩性。这些研究有望为后续开发和实现一大批新的算子方程离散化方法奠定基础，这些方法在处理诸如反应扩散方程、线性输运方程或具有不连续扩散系数的椭圆偏微分方程等一类重要问题时，优于当前基于小波或有限元的方法。

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.