Fast and Reliable Hierarchical Structured Methods for More General Matrix Computations

Purdue University, West Lafayette IN

Investigators

Abstract

Large matrix computations play a critical role in modern scientific computing tasks and engineering simulations. Realistic computations usually involve enormous amounts of data due to large dense matrices or dense intermediate matrix blocks, which makes classical matrix methods impractical. Hierarchical structured methods provide an effective and reliable way to compress and process large matrix data. In such methods, dense matrix blocks are approximated by compact structured forms that are convenient to handle. This research project aims to develop theoretical foundations for understanding multiple hierarchical structured techniques and for designing new hierarchical structured algorithms. These algorithms are expected to be applicable to more general matrix computations and challenging applications where usual structured methods are not suitable or effective. Hierarchical structured methods exploit inherent structures in matrix computations to gain high efficiency while ensuring superior stability. This project is concerned with the design, analysis, and application of fast and reliable hierarchical structured methods for broad classes of challenging computations. A unified framework will be provided to understand multiple types of hierarchical structured methods, design new hierarchical methods with enhanced applicability, and analyze their accuracy and stability. State-of-the-art fast and stable solvers will be developed for tackling challenges such as large data sizes, ill conditioning, high frequencies, and multiple frequencies. The new solvers will be applicable to a wide range of matrix computations. Based on data sparsity and enhanced stability, the solvers will significantly improve the efficiency and reliability of many computations in PDE solution, large data analysis, network, machine learning, imaging, seismic modeling, electromagnetics, etc. The research will also make fast and stable structured solvers widely accessible to broader fields and industries. The data will be included in data repositories for unrestricted access. Open-source packages and educational/tutorial materials will be freely available. The multidisciplinary project will provide excellent opportunities for graduate and undergraduate students from diverse backgrounds to closely interact and to learn critical computational and mathematical skills from multiple fields. 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.