Efficient Sructured Direct Solvers and Robust Structured Preconditioners for Large Linear Systems and Their Applications

Purdue University, West Lafayette IN

Investigators

Abstract

In this project, the investigator and his students design new efficient structured matrix techniques for large linear systems, including fast direct solvers and robust effective preconditioners. These techniques take advantage of certain hidden rank structures in linear systems arising from practical applications. Efficient multi-layer structures and flexible rank requirements are considered. The methods have nearly linear complexity for linear systems arising from the discretization of certain partial differential equations. They can also work as effective preconditioners. Robustness of preconditioning for positive definite problems is shown. The methods are useful for problems which have been considered difficult for classical direct or iterative solvers. This project has broader impacts in many complex numerical problems and engineering simulations, such as differential equations, seismic imaging, climate, electromagnetic field simulation, signal processing, and integrated circuit simulation. The major computational work in these applications is often to solve large-scale linear systems, which can benefit from the efficient black-box solvers or preconditioners developed in this project. These methods help break some classical lower complexity bounds. Students are involved in all aspects of the project, and are trained in various mathematical and engineering areas. The investigator's team plan to build a freely available open source package for both practical applications and education. Minisymposia organized by the investigator, as well as conferences talks, seminars, and journal articles, are used to exchange ideas and to disseminate the results.