Applications of frames to problems in mathematics and engineering

框架在数学和工程问题中的应用

• 批准号: 1008183
• 负责人: Peter Casazza
• 金额: $ 36.9万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1008183&HistoricalAwards=false
• 关键词: Applications; frames; problems; mathematics; engineering

CasazzaDMS-1008183 The investigator works on several fundamental questionsconcerning applications of Hilbert space frames. Much of thework concerns "fusion frames," which provide a natural frameworkfor performing hierarchical data processing. Recent advances inhardware technology have enabled the economic production anddeployment of a large number of low-cost components, which incombination enable reliable and efficient operation. Across manydisciplines there is a fundamental shift from centralizedinformation processing to distributed or network-wide informationprocessing. Data communication is shifting from point-to-pointcommunication to packet transport over wide area networks wherenetwork management is distributed and the reliability of individuallinks is less critical. Radar imaging is moving away from singleplatforms to multiple platforms that cooperate to achieve betterperformance. Wireless sensor networks are emerging as a newtechnology that potentially enables cost-effective and reliablesurveillance. All these applications involve a large number of datastreams, which need to be integrated at a central processor. Fusionframes are a recent development designed precisely for theseapplications. The investigator develops concrete constructions offusion frames and the algorithms needed for their implementation. Working directly with engineers guarantees that advances here haveImmediate implementation. The techniques are also applied toproduce Grassmannian frames for quantum physics -- especiallyQuantum information theory, quantum state tomography, and quantumcryptography. Finally, frame theory has shown that the 1959Kadison-Singer Problem is equivalent to fundamental unsolved problemsin a dozen areas of research in pure mathematics, applied mathematics,and engineering. This project develops the needed frames for resolvingwhat is now one of the most significant open problems in mathematics. We have now entered a new era where the Defense Department,Homeland Security, and industry must process amounts of data thatoverwhelm even current computer capacities. Especially fordefense, it is critical to be able to process these data in realtime, which is becoming more and more difficult. The problemscaused by this "information overload" are escalating rapidlyevery year and need to be addressed immediately and quickly. Therefore, it is of vital national interest to automate as muchof the information fusion process as possible. But at this time,in the area of information fusion the fundamental mathematicsnecessary for such an automated system is far behind the currentneeds. The investigator and his students develop the mathematicsnecessary for tackling these critical needs. This includesmethods for storing and processing information in parallel --i.e. dividing it into multiple subsystems for simpler processingand then "fusing" the results to get the desired outcomes. Italso includes developing distributed sensing networks that cankeep an area under surveillance while taking into account severephysical limitations of the systems such as low communicationbandwidth, limited signal processing power, limited battery life,or the topography of the surveillance area. Such an automatedsystem must be able to combine a large number of sources of quitedisparate information while leaving almost no room for error. The investigator and his students address these needs as well astry to anticipate the next level of difficulties that will arise.

CasazzaDMS-1008183 研究人员致力于研究有关希尔伯特空间框架应用的几个基本问​​题。 大部分工作涉及“融合框架”，它提供了执行分层数据处理的自然框架。 硬件技术的最新进展使得大量低成本组件得以经济生产和部署，这些组件的组合实现了可靠和高效的运行。 许多学科都发生了从集中式信息处理向分布式或网络范围内的信息处理的根本性转变。 数据通信正在从点对点通信转向广域网上的数据包传输，在广域网上，网络管理是分布式的，各个链路的可靠性不再那么重要。 雷达成像正在从单一平台转向多个平台，通过合作实现更好的性能。 无线传感器网络正在成为一种新技术，有可能实现经济高效且可靠的监控。 所有这些应用程序都涉及大量数据流，需要在中央处理器上集成。 Fusionframes 是专为这些应用而设计的最新开发成果。 研究人员开发了融合框架的具体结构及其实现所需的算法。 直接与工程师合作可确保此处的进步立即得到实施。 这些技术还应用于产生量子物理学的格拉斯曼框架——特别是量子信息论、量子态断层扫描和量子密码学。 最后，框架理论表明，1959 年的卡迪森-辛格问题相当于纯数学、应用数学和工程学等十几个研究领域中尚未解决的基本问题。 该项目开发了解决目前数学中最重要的开放问题之一所需的框架。 我们现在已经进入了一个新时代，国防部、国土安全部和工业界必须处理大量数据，这些数据甚至超出了当前计算机的处理能力。 特别是对于国防而言，能够实时处理这些数据至关重要，而这变得越来越困难。 这种“信息过载”引起的问题每年都在迅速升级，需要立即、迅速地解决。因此，尽可能多地实现信息融合过程的自动化对于国家利益至关重要。 但目前，在信息融合领域，这种自动化系统所需的基础数学远远落后于当前的需求。 研究者和他的学生开发了解决这些关键需求所需的数学知识。 这包括并行存储和处理信息的方法——即将其划分为多个子系统以进行更简单的处理，然后“融合”结果以获得所需的结果。 它还包括开发分布式传感网络，该网络可以对某个区域进行监视，同时考虑到系统的严重物理限制，例如低通信带宽、有限的信号处理能力、有限的电池寿命或监视区域的地形。 这样的自动化系统必须能够结合大量完全不同的信息源，同时几乎不留任何出错的空间。调查员和他的学生解决这些需求，并尝试预测将出现的下一级别的困难。