Multiscale Modeling and Approximation in Novel Geometric and Nonlinear Settings

多尺度数据表示已被证明是最有效的数据表示方法之一。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（相关的）问题，例如：这些方法何时起作用？当它们坏了如何修复？如何将这些方法应用到新的场景中？提出的项目在新颖的几何和非线性设置中开发各种数据的多尺度表示；这种表示对这些数据的作用就像小波对图像和信号的作用一样。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，这些任务是信息技术（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）、我们分析和合成许多新型数据的目标正好符合寻找有效方法来组织和操纵大量复杂的高维数据的广泛和基本目标。这样的数据分析问题在科学、医学、工程中变得如此普遍和复杂，以至于应用抽象数学技术的需求变得富有成效和不可避免。该项目为研究生提供跨学科的研究和培训机会，并促进计算数学家、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.