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中的基本挑战，包括工业和科学部门，医疗技术和大数据分析中的挑战，其结果具有深远的影响。其主要目标是开发数学见解，使算法的设计能够有效地处理现代技术日益增长的需求，这些技术产生了广泛而复杂的数据集。具体而言，该项目强调复杂波信号的数据稀疏性的调查，旨在确定最佳的数学组件，或“积木”，用于表达和简化数据。通过揭示数据中的潜在模式和结构，该项目旨在实现更准确和更有洞察力的分析。在机构一级，该项目提供了一个宝贵的机会，培训本科生和研究生，将研究活动纳入教学和课程开发，并通过学术和公共渠道广泛传播和公布研究结果。该项目旨在促进STEM领导力的发展、教育课程的改进以及科学追求的公平性和包容性的进步。该项目为期三年，旨在开发降低成本并提高捕捉波现象准确性的计算方法。该项目旨在发现多尺度表示，计算时频分析和鲁棒的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，展示这些想法在数值逼近和信号恢复方面的潜力，该奖项反映了NSF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进行评估，被认为值得支持的搜索.

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 undergraduate students and graduate students, integrate research activities into teaching and course development, and widely disseminate and divulge findings through academic and public channels. 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.