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

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.