Domain Decomposition Accelerators for Robust Krylov Subspace Methods – DARK

This proposal aims to develop efficient, scalable and robust parallel solvers for linear systems that arise in applied sciences and industrial applications. This is an important topic in high performance computing because linear solves are a bottleneck for many simulations. Indeed, if the problem is too large to be solved by a direct solver, the choice of an iterative solver and of its parameters is a fine art.

Three different scopes have been identified as having particular interest for applications and potential for development. The first is the set of symmetric positive definite linear systems that arises when elliptic PDEs with stochastic coefficients are solved by sampling methods. The second is non-symmetric positive definite matrices. In PDE simulation, non-symmetry may arise from: a convective term, a boundary or coupling condition, a multi-resolution discretization scheme... Finally, general non-symmetric linear systems will be considered with particular attention on wave propagation problems.

The method intended here is to develop deflated Krylov subspace methods (conjugate gradients or GMRES) preconditioned by domain decomposition. New techniques for automatically choosing the preconditioner and deflation operator will be proposed. To arrive at this result, the project will add value to theoretical aspects, methodological aspects and applications. The analysis of Krylov subspace methods and domain decomposition will be jointly advanced to prove new convergence bounds. In these new bounds, the impact of the preconditioner and deflation operator will be explicit. In other words, it will be known what makes a certain choice of preconditioner and deflation operator good or bad. Theory will be carried over to solver design by defining new preconditioners and deflation operators that fit the new criterion. Finally, the new solvers will be implemented in PETSc, tested on a predefined set of applications and made available to end-users.