Nonlinear Sampling Theory: Sparsity, Localization and Optimization

最近的技术进步使得重新考虑在空间上分布大量具有驱动、传感、计算和电信功能的小型设备的想法成为可能。这样的空间实现可以为空间分布式网络中的新应用提供前所未有的能力。研究人员探讨了分布式信号采集系统的非线性采样方法，并注重工程应用。该跨学科项目对数学、工程和工业问题具有广泛的影响，从非线性框架理论和非线性信号处理到智能能源、交通和安全网络。学生们在这个项目的过程中得到了训练。在这个项目中，首席研究员开发了一个新的数学框架来分析和合成压缩参数表示的空间信号。他为通信和能源资源有限的情况开发了全球健壮的算法。他研究了工程网络中智能信号采集分析和综合的最佳采样方法的发展。该项目的主要主题是基于空间域的局部化技术，稀疏测度的拟巴纳赫代数松弛，以及巴纳赫空间中拟最优性的hilberization。

Recent technological advances have made it possible to reconsider the idea of distributing spatially a large number of small devices that have actuating, sensing, computing, and telecommunications capabilities. Such spatial implementations can provide unprecedented capabilities for new applications in spatially distributed networks. The investigator explores methods of nonlinear sampling for distributed signal acquisition systems, with attention to engineering applications. This interdisciplinary project has broader impact on mathematical, engineering and industrial problems, ranging from nonlinear frame theory and nonlinear signal processing to smart energy, transportation, and security networks. Students are trained in the course of this project. In this project, the principal investigator develops a novel mathematical framework to analyze and synthesize spatial signals with compressive parametric representations. He develops globally robust algorithms for cases where communication and energy resources are limited. He investigates development of optimal sampling methodologies for analysis and synthesis of smart signal acquisition in engineering networks. The main themes of this project are based on localization techniques in spatial domains, quasi-Banach algebraic relaxations of sparsity measures, and Hilbertization of quasi-optimality in Banach spaces.