Algebraic domain-decomposition preconditioners for the solution of linear systems

The need to solve linear systems of equations is ubiquitous in scientific computing. Powerful methods for preconditioning such systems have been developed in cases where we can exploit knowledge of the origin of the linear system; a recent example from the solution of systems from PDEs is the Gen-EO domain decomposition method which works well, but requires a non-trival amount of knowledge of the underlying problem to implement. In this talk I will present a new spectral coarse space that can be constructed in a fully-algebraic way, in contrast to most existing spectral coarse spaces, and will give a theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient, especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

Tyrone Rees leads the Computational Mathematics Group at the UK's STFC Rutherford Appleton Laboratory. His research interests lie at the intersection of numerical linear algebra, optimization and the numerical solution of PDEs. Tyrone has a passion for developing numerical software that is widely used; he has oversight of the HSL Mathematical Software Library and is a core developer for the RALFit and FitBenchmarking packages.

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