Numerical discretizations of partial differential equations (PDEs) often lead to large-scale sparse linear systems, which can be challenging to solve. While direct methods are accurate and robust for moderately sized problems, preconditioned iterative methods are more promising for larger-scale problems, for which effective and even optimal preconditioners are critical, especially when the matrices are ill-conditioned or nearly singular. To address these challenges, we have developed a new theoretical framework for optimal preconditioners, called epsilon-accurate approximate generalized inverse, and a corresponding preconditioner called Hybrid Incomplete Factorization (HIF). We show that these preconditioners are approximately optimal in terms of minimizing the number of iterations. We then apply our theory and preconditioner to several challenging problems arising from PDEs, including singular systems from linear elasticity with pure Neumann boundary conditions, saddle-point problems from incompressible Navier-Stokes equations, and large-scale systems from time-dependent PDEs with fully implicit Runge-Kutta schemes. With optimal preconditioners, we then shift our focus to optimal iterative solvers. We discuss some open problems and potential solutions, focusing on leveraging the increasingly ubiquitous half-precision arithmetic on emerging computer architectures.

Coffee, tea, and snacks will be served before the talk, starting at 2:30.

IAM/MECH joint seminar.