Hierarchical hybrid sparse linear solver for multicore platforms
Résumé
The solution of large sparse linear systems is a critical operation
for many numerical simulations. To cope with the hierarchical design
of modern supercomputers, hybrid solvers based on Domain Decomposition
Methods (DDM) have been been proposed. Among them, approaches
consisting of solving the problem on the interior of the domains with
a sparse direct method and the problem on their interface with a
preconditioned iterative method applied to the related Schur
Complement have shown an attractive potential as they can combine the
robustness of direct methods and the low memory footprint of iterative
methods. In this report, we consider an additive Schwarz preconditioner
for the Schur Complement, which represents a scalable candidate but
whose numerical robustness may decrease when the number of domains
becomes too large. We thus propose a two-level MPI/thread parallel
approach to control the number of domains and hence the numerical
behaviour. We illustrate our discussion with large-scale matrices
arising from real-life applications and processed on both a modern
cluster and a supercomputer. We show that the resulting method can
process matrices such as tdr455k for which we previously either ran
out of memory on few nodes or failed to converge on a larger number of
nodes. Matrices such as Nachos_4M that could not be correctly processed
in the past can now be efficiently processed up to a very large number
of CPU cores (24,576 cores). The corresponding code has been
incorporated into the MaPHyS package.
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