| dc.description.abstract | Real-world applications require the solution of large a sparse system of algebraic equations that
arise from the discretization of partial di erential equations with the help of supercomputers.
Modern supercomputers are heterogeneous - each node composed of multi-core CPUs and
many-core GPUs. Porting existing sequential applications speci cally to the GPU architecture
has lead to poor utilization of CPU computing power. In this respect, we develop hybrid
parallel smoothers for the geometric multigrid method which is one of the most e cient
solvers for a system of equations. We study the performance of multigrid method in terms
of total execution time by employing di erent hybrid parallel approaches, viz. accelerating
the smoothing operation using only GPU across all multigrid levels, alternately switching
between GPU and CPU based on the multigrid level and our proposed novel approach of
using combination of GPU and CPU across all multigrid levels. The performance of the hybrid
parallel approaches, implemented using MPI, CUDA and OpenMP, is compared against the
MPI only approach.
In the rst part of the work, we have implemented the hybrid parallel approaches for
the Jacobi and Gauss-Seidel smoothers and tested it to solve the system arising from the
discretization of the Poisson equation. A coloring strategy is developed to color the degrees of
freedom (DOFs) such that the independent set of DOFs are assigned the same color and are
updated in parallel on GPU. We adopted two of the commonly used techniques, CSR Scalar
and CSR Vector, that perform sparse matrix-vector multiplication on GPU, to implement the
smoothing iterations. Further, the strong scaling behavior of the hybrid parallel smoothers is
studied across di erent problem sizes, nite element types, standard and multilevel multigrid.
In the second part of the work, we have implemented the hybrid parallel approaches for
Vanka-type smoother which are typically used to solve the saddle-point problem arising from
the Navier-Stokes equation. We studied the time taken by the two operations viz. assembling
and solving of the local systems in cell and nodal Vanka. We have accelerated the operation of
assembling the local system using the hybrid parallel approaches. A similar coloring strategy
is developed to assign the same color to independent cells or pressure DOFs. The task of
determining the neighbors of each cell or DOF is o oaded to GPU as it is an O(N2) operation.
The operation of solving the assembled local systems is parallelized using OpenMP on CPU.
Similar to the rst part, experiments are performed to study the strong scaling results across
di erent problem sizes, number of OpenMP threads, standard and multilevel multigrid.
The experimental results for the di erent smoothers show that the scaling performance of
the hybrid parallel approaches is bounded by the degree of achievable thread parallelism which
in turn is dependent on the parallel workload per process and the algorithm itself. To improve
the scaling behavior, we propose a combination approach that uses a workload heuristic to
decide the best approach to be applied at each level of multigrid. The combination approach
improves the scaling behavior in addition to resulting in a signi cant speedup under appropriate
workload and number of MPI processes compared to the MPI only approach. | en_US |
