  The solution of linear systems of equations is a central task in a number of
scientific and engineering applications. In many cases the solution of linear
systems may take most of the simulation time thus representing a major
bottleneck in the further development of scientific and technical software. For
large scale simulations, nowadays accounting for several millions or even
billions of unknowns, it is quite common to resort to preconditioned iterative
solvers for exploiting their low memory requirements and, at least potential,
parallelism. Approximate inverses have been shown to be robust and effective
preconditioners in various contexts. In this work, we show how adaptive FSAI,
an approximate inverse characterized by a very high degree of parallelism, can
be successfully implemented on a distributed memory computer equipped with GPU
accelerators. Taking advantage of GPUs in adaptive FSAI set-up is not a trivial
task, nevertheless we show through an extensive numerical experimentation how
the proposed approach outperforms more traditional preconditioners and results
in a close-to-ideal behaviour in challenging linear algebra problems.
