Numerical and harmonic analysis of problems with anisotropic features, directional representation systems and the solution of transport dominated problems, in particular, for parameter dependent high dimensional versions

各向异性特征问题的数值和调和分析、方向表示系统以及传输主导问题的解决方案，特别是参数相关的高维版本

• 批准号: 79152622
• 负责人: Professor Dr. Wolfgang Dahmen
• 金额: --
• 依托单位: Seminar für Angewandte Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79152622?language=en
• 关键词: Numerical; harmonic; analysis; problems; anisotropic

Many important multivariate problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. Examples are digital images as well as solutions of hyperbolic conservation laws or more generally of transport dominated equations. Over the past few years substantial progress has been made in efficiently encoding signals with anisotropic features based on new directional representation systems like shearlets. Although these developments for explicitly given signals are far from complete their understanding has by far more matured than analogous considerations for implicitly given objects like solutions to operator equations of the above type. The central objective of this project is therefore the development and understanding of new discretization concepts that are able to economically and reliably capture anisotropic phenomena in solutions governed by anisotropic phenomena. A central issue is finding suitable variational formulations that on one hand support adaptive solution concepts for transport operators and, on the other hand, are suitable for treating parameter dependent and therefore also high dimensional problems.

许多重要的多元问题类是由各向异性的功能，如奇点集中在低维嵌入流形。例子包括数字图像以及双曲守恒定律或更一般的输运主导方程的解。在过去的几年中，已经取得了实质性的进展，有效地编码信号的各向异性功能的基础上新的方向表示系统，如剪切波。虽然显式给定信号的这些发展远未完成，但它们的理解比隐式给定对象（如上述类型的算子方程的解）的类似考虑要成熟得多。因此，该项目的中心目标是开发和理解新的离散化概念，这些概念能够经济可靠地捕获各向异性现象所支配的解决方案中的各向异性现象。一个中心问题是找到合适的变分制剂，一方面支持自适应解决方案的概念，运输运营商，另一方面，是适合于治疗参数依赖，因此也高维问题。