Frame mechanics: Dynamical principles for optimal redundant expansions

框架力学：最佳冗余扩展的动力学原理

• 批准号: 1109545
• 负责人: Bernhard Bodmann
• 金额: $ 21.49万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109545&HistoricalAwards=false
• 关键词: Frame; mechanics; Dynamical; principles; optimal

BodmannDMS-1109545 The concept of frame mechanics addresses the need for constructing an abundance of optimal redundant, stable expansions with frames, which have become central to applications of mathematics in remote sensing or wireless transmissions, in analog-digital conversion such as audio and video encoding, in packet-based network communications, noise-insensitive quantum computing and recently also in compressive sensing. Despite its popularity, the search for near-optimal frames has been successful mostly in small dimensions, or it had to rely on specific group-representation properties, or the use of randomization principles. In frame mechanics, the investigator is studying an alternative to the conventional, structured or random design methods by letting frames evolve under flows which drive them towards optimality, instead of constructing them directly. The general objectives are to find (1) appropriate frame dynamics, (2) suitable initializations, and to obtain (3) deterministic control of the approximation error. The envisioned outcome of the project includes leveraging recently established numerical results on the construction of equiangular tight frames for the verification of Zauner's conjecture (the existence of maximal Gabor frames in all finite-dimensional Hilbert spaces), constructing controlled approximations of Grassmannian frames and fusion frames for loss-insensitive transmissions in wireless or packet-based network communications, and the design of matrices for compressive sensing based on quantum chaotic dynamics which improve the restricted isometry properties of sensing matrices. The mathematics of redundant signal representations is called frame theory. For practical purposes, a frame is a tool which incorporates or removes repetitive information when data is stored, transmitted or received. Frames have become essential in many data-intensive areas of modern technology, because the repetitive information helps compensate errors of transmission devices and sensors. However, over the last decades, progress in the optimal design of frames has been outpaced by the rapid growth of data generated by our hardware. In frame mechanics, the investigator and his students explore a fundamentally new strategy to overcome this problem: The burden of constructing such optimal frames is put on the computer, which lets frames evolve in a way that drives them towards optimality. The goal of this project is to demonstrate that this dynamic design strategy is mathematically guaranteed to find many optimal frames where previous attempts failed. Frame mechanics allows us to maximize performance in remote sensing, seismic and medical imaging, wireless and fiber-optic communications, and to make internet transmissions robust to network outages.

Bodmanndms-11109545框架机制的概念解决了构建丰富的范围冗余，稳定的扩展的需求，这些扩展已成为数学在遥感或无线传输中的应用，以及在基于数据的网络通信中，在远程感应或无线传输中的应用，以及在基于数据的网络通信中，以及最近的噪声构成量和视频传感。 尽管它很受欢迎，但对近乎最佳的框架的搜索主要是在很小的方面成功，或者必须依靠特定的群体代理属性或使用随机原则。 在框架力学中，研究人员正在研究传统，结构化或随机设计方法的替代方案，通过让框架在流动下进化，从而使它们朝着最优性而不是直接构造。 一般目标是找到（1）适当的帧动力学，（2）合适的初始化，并获得（3）对近似误差的确定性控制。 该项目的设想结果包括利用最近确定的等于等距离紧密框架的数值结果，以验证Zauner的猜想（在所有有限维度的希尔伯特空间中存在最大Gabor框架的存在），以构建了跨性别框架和型号的跨性别框架，并构建了跨性别框架的近似值，并构建了损失框架的近似值。基于量子混沌动力学的压缩感测的矩阵，从而提高了传感矩阵的限制等轴测特性。 冗余信号表示的数学称为框架理论。 出于实际目的，框架是一种工具，该工具在数据存储，传输或接收时将重复信息包含或删除。 在现代技术的许多数据密集型领域中，框架已成为必不可少的，因为重复信息有助于补偿传输设备和传感器的错误。 但是，在过去的几十年中，由于硬件生成的数据的快速增长，最佳设计的最佳设计进展已超过。 在框架力学中，研究人员和他的学生探索了一种从根本上克服这个问题的新策略：构建此类最佳框架的负担被放在计算机上，这使得框架以驱动它们朝着最优性驱动的方式发展。 该项目的目的是证明这种动态设计策略在数学上可以保证找到以前尝试失败的许多最佳框架。 框架机制使我们能够在遥感，地震和医学成像，无线和光纤通信中最大程度地提高性能，并使互联网传输可靠地进行网络中断。