In this research effort the investigator creates mathematical concepts and numerical methods for information technology, communications engineering, and data analysis. The investigator uses tools from pseudodifferential operator theory, time-frequency analysis, random matrix theory, and Banach algebra theory, yielding efficient numerical algorithms with rigorously-established properties under carefully stated conditions. Some concrete topics of this research effort are: (i) Development of a theoretical framework for key problems in classical and quantum information theory. Specifically, the investigator considers the channel capacity problem in time-varying communications and quantum communications; (ii) Sparse representations and compressed sensing in X-ray crystallography, communications, and radar. Initial steps toward building a framework for nonlinear compressed sensing; (iii) Noncommutative harmonic analysis and pseudodifferential operators from the point of view of computational efficiency and the development of fast algorithms. Particular attention is paid to spectral factorization for operators in a noncommutative setting, and their application in signal processing and wireless communications. Strong expectation for success of this project can be based on existing solid achievements by the investigator in each of the described areas.
The research proposed in this effort is a marriage of several areas of cutting edge mathematics with state-of-the-art engineering, seeking to bring advanced techniques from abstract and applied harmonic analysis to communications engineering, signal processing, and data analysis in form of fast and efficient computational methods. By taking the modern harmonic analysis methodology into the engineering community this research activity will enable further advances and breakthroughs in important applications. At the same time it will stimulate new research areas in applied mathematics and pave the road for further interactions between applied mathematicians and engineers. The payoffs of this research effort for society at large are many, ranging from new information technology capabilities and sophisticated tools to deal with today's massive volumes of data. There are financial efficiencies to be gained by communications providers which will be accompanied by better and increased communications services for the public. Other potential benefits include improved methods for medical imaging and biomedical engineering. Beyond the project's broad technological impact, it serves as a model for the kind of cross-disciplinary activity critical for research and education at the mathematics/engineering frontier. Hence this research effort helps to train graduate students in mathematics to develop and enhance skills that are crucial and urgently needed in a high-tech oriented society.
This research effort has created mathematical concepts and numerical methods for information technology, communications engineering, and data analysis. The carried out research is a marriage of several areas of cutting edge mathematics with state-of-the-art engineering, and has brought advanced techniques from abstract and applied harmonic analysis to communications engineering, signal processing, and data analysis in form of fast and efficient computational methods. In recent years Compressive Sensing, a fascinating new area at the intersection of mathematics, computer science, and electrical engineering, has attracted enormous attention from academia and industry. The core idea is that data acquisition and compression can be performed simultaneously. Several topics of this research project are related to compressive sensing. Among others, this project focused on: (i) compressive sensing radar, (ii) high-dimensional data analysis and compressive measurements, (iii) sparsity in wireless communications, and (iv) and nonlinear compressive sensing. To describe just a few concrete result, the mathematical research of this project paves the way to the design of new radar systems with hitherto unprecedented capabilities, such as the ability to detect multiple and potentially rather small targets at very high resolution in the azimuth/range/Doppler domain. Furthermore, the research on sparse representations has led to crucial new insight into the connection between Inverse Problems, Compressive Sensing, Random Matrices, and Regularization Theory. Another major achievement is the development of a new approach to phase retrieval, which is a central problem in many scientific disciplines (ranging from X-ray crystallography to quantum tomography and Terahertz imaging) and notoriously difficult to solve. This project has produced conceptual deliverables in the form of new mathematical methods for information theory, inverse scattering, and data analysis. The project has also produced concrete deliverables in the form of numerical algorithms for various real-world applications for use in the scientific and industrial sector. Beyond the project's broad technological impact, it serves as a model for the kind of cross-disciplinary activity critical for research and education at the mathematics/engineering frontier. This research model has been propagated through the graduate students trained with the grant, as well as via the classroom, through the teaching of graduate courses that emphasized the interdisciplinary nature of this project. Hence this research effort has helped to train graduate students in mathematics to develop and enhance skills that are crucial and urgently needed in our high-tech oriented society.
