Sparse direct solvers play a vital role in large-scale high performance computing in science and engineering. Existing distributed sparse direct methods employ multifrontal/supernodal patterns to aggregate columns of nearly identical forms and to exploit dense basic linear algebra subprograms (BLAS) for computation. However, such a data layout may bring more unevenness when the structure of the input matrix is not ideal, and using dense BLAS may waste many floating-point operations on zero fill-ins. In this paper, we propose a new sparse direct solver called PanguLU. Unlike the multifrontal/supernodal layout, our work relies on simpler regular 2D blocking and stores the blocks in their sparse forms to avoid any extra fill-ins. Based on the sparse patterns of the blocks, a variety of block-wise sparse BLAS methods are developed and selected for higher efficiency on local GPUs. To make PanguLU more scalable, we also adjust the mapping of blocks to processes for overall more balanced workload, and propose a synchronisation-free communication strategy considering the dependencies among different sub-tasks to reduce overall latency overhead. Experiments on two distributed heterogeneous platforms consisting of 128 NVIDIA A100 GPUs and 128 AMD MI50 GPUs demonstrate that PanguLU achieves up to 11.70x and 17.97x speedups over the latest SuperLU_DIST, and scales up to 47.51x and 74.84x on the 128 A100 and MI50 GPUs over a single GPU, respectively.

PanguLU: A Scalable Regular Two-Dimensional Block-Cyclic Sparse Direct Solver on Distributed Heterogeneous Systems

The downside of increasing the resolution of surface scanning devices is that the amount of acquired data makes the morphological analysis of the scanned surfaces computationally challenging. This limitation is circumvented by using scalable and non–memory–intensive harmonic expansions. In this paper, the projection of parametric surfaces onto disk and spherical harmonics bases is investigated, and three novel computationally efficient algorithms are proposed based on the randomised Kaczmarz (RK). To boost the computational performance and convergence of the root mean square error (RMSE) of the reconstructed surfaces we exploited the conjugate symmetry property of the harmonic basis functions. Further, the sparsity of the signals is used for estimating the projection coefficients from an undersampled surface. The first algorithm only takes into consideration the conjugate symmetry property for enhancing the convergence of the RMSE. The second algorithm endows the sparse version of the RK algorithm with the conjugate symmetry property. The third algorithm combines the previous two to further accelerate the convergence of the RMSE. The performance of the developed algorithms is tested on three surfaces where we demonstrate that they outperform conventional reconstruction techniques in terms of processing time with comparable precision. 

On the conjugate symmetry and sparsity of the harmonic decomposition of parametric surfaces with the randomised Kaczmarz method

Sparse matrix vector multiplication (SpMV) is a critical performance bottleneck for numerical simulation and artificial intelligence training. The new generation of Sunway supercomputer is the advanced exascale supercomputer in China. The SW26010-Pro many-core processor renders itself as a competitive candidate for its attractive computational power in both numerical simulation and artificial intelligence training. In this paper, we propose a heterogeneous and SPM-aggregated SpMV kernel, specifically designed for the SW26010-Pro many-core processor. To fully exploit the computational power of the SW26010-Pro and balance the load of each core group(CG) during computation, we employ asynchronous computation workflow and propose the SPM-aggregated strategy and vector adaptive mapping algorithm. In addition, we propose the two-level data partition scheme to implement computational load balance. In order to improve memory access efficiency, we directly access memory via DMA controller to replace the discrete memory access. Using several optimizations, we achieve a 77.16x speedup compared to the original implementation. Our experimental results show that the hsSpMV yields up to 3.82× speedups on average compared to the SpMV kernel of the state-of-the-art Sunway math library xMath2.0. 

hsSpMV: A Heterogeneous and SPM-aggregated SpMV for SW26010-Pro many-core processor

We present sketched linear discriminant analysis, an iterative randomized approach to binary-class Gaussian model linear discriminant analysis (LDA) for very large data. We harness a least squares formulation and mobilize the stochastic gradient descent framework. Therefore, we obtain a randomized classiﬁer with performance that is very comparable to that of full data LDA while requiring access to only one row of the training data at a time. We present convergence guarantees for the sketched predictions on new data within a ﬁxed number of iterations. These guarantees account for both the Gaussian modeling assumptions on the data and algorithmic randomness from the sketching procedure. Finally, we demonstrate performance with varying step-sizes and numbers of iterations. Our numerical experiments demonstrate that sketched LDA can oﬀer a very viable alternative to full data LDA when the data may be too large for full data analysis.

pdf/sketched-gaussian-model-linear-discriminant-analysis-via-the-3tjwk3v2.pdf

Sketched Gaussian Model Linear Discriminant Analysis via the Randomized Kaczmarz Method

swPTS: an efficient parallel Thomas split algorithm for tridiagonal systems on Sunway manycore processors

swSuperLU: A highly scalable sparse direct solver on Sunway manycore architecture

This paper proposes a distributed-memory parallel randomized iterative algorithm for solving linear systems, called the parallel randomized kaczmarz projection (PRKP) algorithm. The algorithm has the property of greedy sampling, alternating projection, and lazy approximation. We derive the alternating projection process from the Randomized Kaczmarz algorithm and develop the greedy sampling and lazy approximation process to improve the convergence rate and reduce the per-process communication volume. Moreover, we develop a sampling residual estimation scheme for our proposed algorithm, which greatly reduces the extra computation cost required to obtain residuals. Our experimental results show that the proposed algorithm significantly outperforms previous parallel iterative algorithms on overdetermined linear systems, and yields up to $\mathbf{159}\times$ and $\mathbf{384}\times$ speedups on average respectively compared to the leading pipelined CG and pipelined GMRES algorithms.

PRKP: A Parallel Randomized Iterative Algorithm for Solving Linear Systems

SPM-GCN: An adaptive reordering algorithm for sparse LU factorization via GCN

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swSuperLU: A highly scalable sparse direct solver on Sunway manycore architecture

PRKP: A Parallel Randomized Iterative Algorithm for Solving Linear Systems

SPM-GCN: An adaptive reordering algorithm for sparse LU factorization via GCN