Exploiting Data-Sparse Structures in Numerical Linear Algebra
Exploring data-sparse structures provides a ubiquitous way to design fast algorithms in
numerical linear algebra. In the first part of this talk, I will show how to identify and take advantage of these structures to develop fast and stable linear system solvers. These solvers have been applied to solve problems arising from structured matrices, integral equations, and PDEs. In the second part, I will present several methods to efficiently extract these structures. These methods have been successfully used in electronic structure calculations and earth mantle simulations. I will also show how to adapt these techniques to build a preconditioner for solving highly indefinite sparse linear systems, with an emphasis on high frequency 3D Helmholtz equations.