Nonlinear Sampling Theory: Sparsity, Localization and Optimization

最近的技术进步使得有可能重新考虑空间分布大量具有致动、感测、计算和电信能力的小型设备的想法。 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。 研究者探讨分布式信号采集系统的非线性采样方法，并关注工程应用。 这个跨学科项目对数学，工程和工业问题产生了更广泛的影响，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。 学生在这个项目的过程中接受培训。 在这个项目中，主要研究者开发了一种新的数学框架来分析和合成具有压缩参数表示的空间信号。 他为通信和能源有限的情况开发了全球强大的算法。 他研究了工程网络中智能信号采集的分析和合成的最佳采样方法的发展。 这个项目的主要主题是基于空间域中的本地化技术，稀疏性措施的准Banach代数松弛，以及Banach空间中准最优性的希尔伯特化。

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.