Publications

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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.

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This article describes a parallel implementation of a two-level overlapping Schwarz preconditioner with the GDSW (Generalized Dryja–Smith–Widlund) coarse space described in previous work [12, 10, 15] into the Trilinos framework; cf. [16]. The software is a significant improvement of a previous implementation [12]; see Sec. 4 for results on the improved performance.

As computer architectures are rapidly evolving (e.g. those designed for exascale), multiple portability frameworks have been developed to avoid new architecture-specific development and tuning. However, portability frameworks depend on compilers for auto-vectorization and may lack support for explicit vectorization on heterogeneous platforms. Alternatively, programmers can use intrinsics-based primitives to achieve more efficient vectorization, but the lack of a gpu back-end for these primitives makes such code non-portable. A unified, portable, Single Instruction Multiple Data (simd) primitive proposed in this work, allows intrinsics-based vectorization on cpus and many-core architectures such as Intel Knights Landing (knl), and also facilitates Single Instruction Multiple Threads (simt) based execution on gpus. This unified primitive, coupled with the Kokkos portability ecosystem, makes it possible to develop explicitly vectorized code, which is portable across heterogeneous platforms. The new simd primitive is used on different architectures to test the performance boost against hard-to-auto-vectorize baseline, to measure the overhead against efficiently vectroized baseline, and to evaluate the new feature called the “logical vector length” (lvl). The simd primitive provides portability across cpus and gpus without any performance degradation being observed experimentally.

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Community detection in graphs is a canonical social network analysis method. We consider the problem of generating suites of teras-cale synthetic social networks to compare the solution quality of parallel community-detection methods. The standard method, based on the graph generator of Lancichinetti, Fortunato, and Radicchi (LFR), has been used extensively for modest-scale graphs, but has inherent scalability limitations. We provide an alternative, based on the scalable Block Two-Level Erdos-Renyi (BTER) graph generator, that enables HPC-scale evaluation of solution quality in the style of LFR. Our approach varies community coherence, and retains other important properties. Our methods can scale real-world networks, e.g., to create a version of the Friendster network that is 512 times larger. With BTER's inherent scalability, we can generate a 15-terabyte graph (4.6B vertices, 925B edges) in just over one minute. We demonstrate our capability by showing that label-propagation community-detection algorithm can be strong-scaled with negligible solution-quality loss.

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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.

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In this report, we abstract eleven papers published during the project and describe preliminary unpublished results that warrant follow-up work. The topic is multi-level memory algorithmics, or how to effectively use multiple layers of main memory. Modern compute nodes all have this feature in some form.

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We present a new method for reducing parallel applications’ communication time by mapping their MPI tasks to processors in a way that lowers the distance messages travel and the amount of congestion in the network. Assuming geometric proximity among the tasks is a good approximation of their communication interdependence, we use a geometric partitioning algorithm to order both the tasks and the processors, assigning task parts to the corresponding processor parts. In this way, interdependent tasks are assigned to “nearby” cores in the network. We also present a number of algorithmic optimizations that exploit specific features of the network or application to further improve the quality of the mapping. We specifically address the case of sparse node allocation, where the nodes assigned to a job are not necessarily located in a contiguous block nor within close proximity to each other in the network. However, our methods generalize to contiguous allocations as well, and results are shown for both contiguous and non-contiguous allocations. We show that, for the structured finite difference mini-application MiniGhost, our mapping methods reduced communication time up to 75 percent relative to MiniGhost’s default mapping on 128K cores of a Cray XK7 with sparse allocation. For the atmospheric modeling code E3SM/HOMME, our methods reduced communication time up to 31% on 16K cores of an IBM BlueGene/Q with contiguous allocation.

Triangle counting is a foundational graph-analysis kernel in network science. It has also been one of the challenge problems for the 'Static Graph Challenge'. In this work, we propose a novel, hybrid, parallel triangle counting algorithm based on its linear algebra formulation. Our framework uses MPI and Cilk to exploit the benefits of distributed-memory and shared-memory parallelism, respectively. The problem is partitioned among MPI processes using a two-dimensional (2D) Cartesian block partitioning. One-dimensional (1D) rowwise partitioning is used within the Cartesian blocks for shared-memory parallelism using the Cilk programming model. Besides exhibiting very good strong scaling behavior in almost all tested graphs, our algorithm achieves the fastest time on the 1.4B edge real-world twitter graph, which is 3.217 seconds, on 1,092 cores. In comparison to past distributed-memory parallel winners of the graph challenge, we demonstrate a speed up of 2.7× on this twitter graph. This is also the fastest time reported for parallel triangle counting on the twitter graph when the graph is not replicated.