M15: Preconditioning non-symmetric systems for high performance computing (HPC)
| Researcher: | Dr Jennifer Pestana |
| Team Leader: | Dr Andy Wathen |
| Collaborators: |
Prof. David Keyes, KAUST |
Background
The solution of high dimensional sets of linear equations
remains the limiting bottle-neck in much of scientific computing. The optimal
and near-optimal complexity algorithms necessary for very large scale problems
in high performance computing (HPC) often rely on rapidly convergent iterative
methods. For nonsymmetric problems, such as those arising from the
Navier–Stokes equations, iterative methods exist, but only practical experience
or luck result in preconditioners that give rapid convergence. It remains a
fundamental and largely open mathematical question as to what one is trying to
achieve with preconditioning for nonsymmetric matrices.
Techniques and Challenges
This aim of this project is to provide a rigorous mathematical framework in which criteria for selecting preconditioners for nonsymmetric matrices that guarantee acceptably rapid convergence.
Results
Many problems possess a structure that can be exploited. For example, we have observed that effective preconditioners for the iterative method GMRES are self-adjoint with respect to a symmetric bilinear form that is close (in some sense) to that which with respect to the coefficient matrix is self-adjoint. This forms a starting point for analysis of preconditioners in general, and in particular for the Pressure-Convection-Diffusion (PCD) and Least-Squares-Commutator (LSC) preconditioners for the steady incompressible Navier-Stokes equations.
The Future
Specifically, we aim to derive mathematical criteria for preconditioners for nonsymmetric matrix equations that guarantee rapid convergence of the GMRES method. We hope to apply these criteria to the PCD and LSC preconditioners for the steady incompressible Navier-Stokes equations.
References
Elman H., Silvester D., Wathen A.: Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics, Oxford University Press, 2005
Pestana J.: Nonstandard Inner Products and Preconditioned Iterative Methods, DPhil Thesis, University of Oxford, 2011