Antz, Hartwig A.; Boman, Erik G.; Gates, Mark G.; Kruger, Scott E.; Li, Sherry L.; Loe, Jennifer A.; Osei-Kuffuor, Daniel O.; Tomov, Stan T.; Tsai, Yaohung M.; Meier Yang, Ulrike M.
The use of multiple types of precision in mathematical software has the potential to increase its performance on new heterogeneous architectures. The xSDK project focuses both on the investigation and development of multiprecision algorithms as well as their inclusion into xSDK member libraries. This report summarizes current efforts on including and/or using mixed precision capabilities in the math libraries Ginkgo, heFFTe, hypre, MAGMA, PETSc/TAO, SLATE, SuperLU, and Trilinos, including KokkosKernels. It contains both numerical results from libraries that already provide mixed precision capabilities, as well as descriptions of the strategies to incorporate multiprecision into established libraries.
Proceedings of IA3 2020: 10th Workshop on Irregular Applications: Architectures and Algorithms, Held in conjunction with SC 2020: The International Conference for High Performance Computing, Networking, Storage and Analysis
Graph coloring is often used in parallelizing scientific computations that run in distributed and multi-GPU environments; it identifies sets of independent data that can be updated in parallel. Many algorithms exist for graph coloring on a single GPU or in distributed memory, but hybrid MPI+GPU algorithms have been unexplored until this work, to the best of our knowledge. We present several MPI+GPU coloring approaches that use implementations of the distributed coloring algorithms of Gebremedhin et al. and the shared-memory algorithms of Deveci et al. The on-node parallel coloring uses implementations in KokkosKernels, which provide parallelization for both multicore CPUs and GPUs. We further extend our approaches to solve for distance-2 coloring, giving the first known distributed and multi-GPU algorithm for this problem. In addition, we propose novel methods to reduce communication in distributed graph coloring. Our experiments show that our approaches operate efficiently on inputs too large to fit on a single GPU and scale up to graphs with 76.7 billion edges running on 128 GPUs.
Graph partitioning has been an important tool to partition the work among several processors to minimize the communication cost and balance the workload. While accelerator-based supercomputers are emerging to be the standard, the use of graph partitioning becomes even more important as applications are rapidly moving to these architectures. However, there is no scalable, distributed-memory, multi-GPU graph partitioner available for applications. We developed a spectral graph partitioner, Sphynx, using the portable, accelerator-friendly stack of the Trilinos framework. We use Sphnyx to systematically evaluate the various algorithmic choices in spectral partitioning with a focus on GPU performance. We perform those evaluations on irregular graphs, because state-of-the-art partitioners have the most difficulty on them. We demonstrate that Sphynx is up to 17x faster on GPUs compared to the case on CPUs, and up to 580x faster compared to a state-of-the-art multilevel partitioner. Sphynx provides a robust alternative for applications looking for a GPU-based partitioner.
We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification, and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly \scrO (N), and the number of iterations grows slowly with N.
A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using 1024 processors.