Collaborative Research: New perspectives from applied and computational time-frequency analysis

该项目探索新的理论和算法，以解决具有重大社会影响的时频分析的基本挑战，包括工业和科学部门，医疗技术和大数据分析，其结果具有深远的影响。主要目标是发展数学洞察力，使算法设计能够有效地处理产生广泛和复杂数据集的现代技术日益增长的需求。具体来说，该项目强调对复杂波信号的数据稀疏性的研究，旨在确定最佳的数学成分，或“构建块”，用于表达和简化数据。通过揭示数据中的底层模式和结构，该项目旨在实现更准确、更深刻的分析。在机构层面，本项目提供了一个宝贵的机会，培养至少两名本科生和两名研究生，将研究活动融入教学和课程开发，并通过学术和公共渠道广泛传播和泄露研究成果。此外，调查人员还坚定地致力于促进STEM领域的公平和包容，举办公开讲座以增加信息泄露，为学生提供指导，并协调外展项目。该项目促进了STEM领导力的发展，提高了教育课程，促进了科学追求的公平性和包容性。这个为期三年的项目旨在开发计算方法，以降低成本并提高捕获波浪现象的准确性。该项目旨在发现多尺度表示、计算时频分析和鲁棒高维采样策略中的稀疏性。通过结合框架理论、线性代数、信息理论和计算科学的见解，该项目旨在开发数值算法，提高计算和存储的速度、精度和成本效益。研究目标包括构建稀疏时频表示，开发高维信号的最佳采样策略，展示这些思想在数值逼近和信号恢复方面的潜力，以及研究应用谐波分析中尚未解决的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project explores new theories and algorithms that address fundamental challenges in time-frequency analysis with significant societal implications, including those in industrial and scientific sectors, medical technology, and big data analytics where the outcomes have far-reaching effects. The primary objective is to develop mathematical insights that enable the design of algorithms capable of efficiently handling the increasing demands of modern technologies which generate extensive and intricate datasets. Specifically, the project emphasizes the investigation of data sparsity of complex wave signals aiming to identify the optimal mathematical components, or "building blocks," for expressing and simplifying the data. By uncovering the underlying patterns and structures within the data, the project aims to enable more accurate and insightful analysis. At an institutional level this project offers a valuable opportunity to train at least two undergraduate students and two graduate students, integrate research activities into teaching and course development, and widely disseminate and divulge findings through academic and public channels. Additionally, the investigators maintain a strong commitment to promoting equity and inclusion in STEM fields, delivering public lectures to increase divulgation, student mentoring, and coordinated outreach programs. This project enables STEM leadership development, enhancements in educational curricula, and advancements in equity and inclusion in scientific pursuits.This three-year project aims to develop computational methods that reduce the cost and improve the accuracy of capturing wave phenomena. The project seeks to discover sparsity in multi-scale representations, computational time-frequency analysis, and robust, high-dimensional sampling strategies. By combining frame theory, linear algebra, information theory, and insights from computational science, the project aims to develop numerical algorithms that enhance speed, precision, and cost-efficiency in computation and storage. The research objectives include constructing sparse time-frequency representations, developing optimal sampling strategies for high-dimensional signals, demonstrating the potential of these ideas for numerical approximation and signal recovery, and investigating unresolved problems in applied harmonic analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.