  Robust iterative methods for solving large sparse systems of linear algebraic
equations often suffer from the problem of optimizing the corresponding tuning
parameters. To improve the performance of the problem of interest, specific
parameter tuning is required, which in practice can be a time-consuming and
tedious task. This paper proposes an optimization algorithm for tuning the
numerical method parameters. The algorithm combines the evolution strategy with
the pre-trained neural network used to filter the individuals when constructing
the new generation. The proposed coupling of two optimization approaches allows
to integrate the adaptivity properties of the evolution strategy with a priori
knowledge realized by the neural network. The use of the neural network as a
preliminary filter allows for significant weakening of the prediction accuracy
requirements and reusing the pre-trained network with a wide range of linear
systems.
  The detailed algorithm efficiency evaluation is performed for a set of model
linear systems, including the ones from the SuiteSparse Matrix Collection and
the systems from the turbulent flow simulations. The obtained results show that
the pre-trained neural network can be effectively reused to optimize parameters
for various linear systems, and a significant speedup in the calculations can
be achieved at the cost of about 100 trial solves. The hybrid evolution
strategy decreases the calculation time by more than 6 times for the black box
matrices from the SuiteSparse Matrix Collection and by a factor of 1.4-2 for
the sequence of linear systems when modeling turbulent flows. This results in a
speedup of up to 1.8 times for the turbulent flow simulations performed in the
paper.
