Jie Liu

Shengguo Li, Francois-Henry Rouet, Jie Liu et al.

In this paper, an efficient divide-and-conquer (DC) algorithm is proposed for the symmetric tridiagonal matrices based on ScaLAPACK and the hierarchically semiseparable (HSS) matrices. HSS is an important type of rank-structured matrices.Most time of the DC algorithm is cost by computing the eigenvectors via the matrix-matrix multiplications (MMM). In our parallel hybrid DC (PHDC) algorithm, MMM is accelerated by using the HSS matrix techniques when the intermediate matrix is large. All the HSS algorithms are done via the package STRUMPACK. PHDC has been tested by using many different matrices. Compared with the DC implementation in MKL, PHDC can be faster for some matrices with few deflations when using hundreds of processes. However, the gains decrease as the number of processes increases. The comparisons of PHDC with ELPA (the Eigenvalue soLvers for Petascale Applications library) are similar. PHDC is usually slower than MKL and ELPA when using 300 or more processes on Tianhe-2 supercomputer.

Jie Liu, Zhongyuan Zhao, Zijian Ding et al.

The ongoing trend of hardware specialization has led to a growing use of custom data formats when processing sparse workloads, which are typically memory-bound. These formats facilitate optimized software/hardware implementations by utilizing sparsity pattern- or target-aware data structures and layouts to enhance memory access latency and bandwidth utilization. However, existing sparse tensor programming models and compilers offer little or no support for productively customizing the sparse formats. Additionally, because these frameworks represent formats using a limited set of per-dimension attributes, they lack the flexibility to accommodate numerous new variations of custom sparse data structures and layouts. To overcome this deficiency, we propose UniSparse, an intermediate language that provides a unified abstraction for representing and customizing sparse formats. Unlike the existing attribute-based frameworks, UniSparse decouples the logical representation of the sparse tensor (i.e., the data structure) from its low-level memory layout, enabling the customization of both. As a result, a rich set of format customizations can be succinctly expressed in a small set of well-defined query, mutation, and layout primitives. We also develop a compiler leveraging the MLIR infrastructure, which supports adaptive customization of formats, and automatic code generation of format conversion and compute operations for heterogeneous architectures. We demonstrate the efficacy of our approach through experiments running commonly-used sparse linear algebra operations with specialized formats on multiple different hardware targets, including an Intel CPU, an NVIDIA GPU, an AMD Xilinx FPGA, and a simulated processing-in-memory (PIM) device.