Integration of Randomized Methods and Fast and Reliable Matrix Computations

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In modern scientific computing, engineering simulations, and data analysis, the complexity of numerical problems and the scale of data sizes pose unique challenges to matrix computations. The demand for efficiency and reliability continues to grow, and in the meantime, researchers increasingly desire algorithms that are convenient to use. Randomized algorithms are not only convenient and fast to apply, but also powerful in the sense that they can sometimes extract surprisingly useful information from challenging situations that are otherwise very difficult to handle. This project will integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms so as to build a comprehensive framework for fast, reliable, and flexible randomized matrix computations. The project will bridge the gap between convenient randomized ideas and various challenging numerical tasks. Innovative randomized methods will be designed to extract valuable information in numerical computations and also to guide algorithm design and parameter tuning. The project will help effectively make randomized strategies more widely accessible to broader scientific communities. It will help introduce novel randomized algorithms into various numerical analysis fields and can also significantly improve the efficiency and reliability of many practical computational tasks in application fields such as data science, image processing, geosciences, and engineering. The research results will be widely disseminated via multiple channels. The project will give students a nice platform to gain knowledge in different areas such as numerical analysis, statistics, and data analysis. Relevant course materials will be developed. Open-source software packages will be designed.The research will seamlessly integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms. The research will result in a series of novel randomized methods for computations such as low-rank approximation, data-sparse preconditioning, and eigenvalue solution. Rigorous theories will be given to understand the effectiveness of the proposed methods and also to establish connections between randomized algorithms and various challenging numerical tasks. Unlike the usual compromise between efficiency and reliability in many randomized strategies, the integration of multiple randomized techniques and novel matrix computations can achieve the combined benefits of flexibility, convenience, efficiency, and also reliability. The studies can further help uncover intrinsic matrix properties that can be used to design robust numerical algorithms for handling difficult situations. The new analysis and randomized methods will help make matrix computations better meet the rapidly emerging challenges of modern computational tasks.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.