Multiscale Modeling and Approximation in Novel Geometric and Nonlinear Settings

多尺度数据表示已被证明是表示数据的最有效方法之一。这些方法不仅在应用数学中，而且在计算机科学和工程(特别是计算机图形学和科学模拟社区)中都是当前的主要兴趣，应用数学家的工作是回答(相互关联的)问题：这些方法什么时候起作用？当它们失效时如何修复它们？如何将这些方法应用到新颖的环境中？拟议的项目开发了在新的几何和非线性环境中对数据的各种多尺度表示；这种表示对这些数据的作用与小波对图像和信号的作用相同。由此产生的多尺度表示是数据压缩、特征提取、噪声去除和许多其他信号处理任务的关键，这些任务对于信息技术(计算机图形学、计算机辅助设计、无线通信等)、医学成像技术(MRI和其他类型的放射学)、军事信号处理(声纳和雷达等)来说是关键。我们分析和合成许多新型数据的目标恰好符合找到有效的方法来组织和处理海量和复杂的高维数据的广泛而基本的目标。这种数据分析问题在整个科学、医学、该项目为研究生提供了跨学科的研究和培训机会，并促进了计算数学家、工程师和科学家之间的合作。

Multiscale data representation has been proven to be one of the most effective methods for representing data. Such methods are of major current interests not only in applied mathematics but also in computer science and engineering (especially the computer graphics and scientific simulation communities), and it is the job of applied mathematicians to answer (interrelated) questions such as: When do these methods work?How to fix them when they break? How to bring these methods to novel settings ? etc..The proposed projects develop various multiscale representations of data in novel geometric and nonlinear settings; such representations do for such data what wavelets were able to do for images and signals. The resulted multiscale representations are the key to data compression, feature extraction, noise removal and a number of other signal processing tasks that are key to informational technologies (computer graphics, computer-aided design, wireless communication, etc..), medical imaging technology (MRI and other kinds of radiology), military signal processing(sonar and radar etc.)Our goal of analysis and synthesis of many new types of data fits right into the broad and fundamental goal of finding efficient ways to organize and manipulate enormous and complex volumes of high-dimensional data.Such data analysis problems have gotten so ubiquitous and sophisticated throughout science, medicine, engineering that the need of applying abstract mathematical techniques becomes fruitful and inevitable.The project provides interdisciplinary research and training opportunities for graduate students, and stimulates collaboration among computational mathematicians, engineers and scientists.