Exploiting matrix symmetry to halve memory footprint offers a substantial opportunity for accelerating memory-bound computations like Sparse Matrix-Vector Multiplication (SpMV). However, symmetric SpMV incurs data conflicts when concurrently writing the output vector. Previous approaches fail to address this issue efficiently, i.e., either are non-scalable or yield poor performance for large high-bandwidth irregular matrices. This article extends DCS-SpMV , a D ivide-and- C onquer (DC) based shared-memory implementation of S ymmetric SpMV. The key idea of DCS-SpMV is to recursively divide and reorder the matrix-induced conflict graph into independent subgraphs for parallel execution, and construct separate subgraphs to avoid data conflicts. The DC approach naturally transforms the input matrix into a low-conflict part and a high-conflict part, which motivates us to design a conflict-aware hybrid solution DCH-SpMV that executes these two parts using DCS-SpMV and the standard SpMV, respectively. We also develop a machine learning model for DCH-SpMV to predict the optimal number of DC recursions on a given matrix and architecture. In this work, we further optimize the hybrid DC implementation by reducing data conflicts before the DC preprocessing. First, we present a conflict-pruning strategy to decouple certain highly dense columns or rows from the conflict graph of a symmetric matrix. Second, we implement a heuristic to adaptively select the lower or upper triangular part of a symmetric matrix, leading to fewer data conflicts. Our optimizations not only facilitate the DC preprocessing, but also improve the performance of DCH-SpMV. We evaluate our work on both x86 and ARM multi-core CPUs using 298 symmetric sparse matrices from the SuiteSparse Matrix Collection. Our new optimizations improve the performance of previous version [ 42 ] by up to 4.89×, demonstrating significant speedup over the state-of-the-art approaches including the vendor-tuned Intel oneMKL library.

This paper focuses on optimizing the Matrix-Power Kernel (MPK), which relies on a series of Sparse Matrix-Vector multiplications (SpMVs) using the same sparse matrix. MPK is a crucial component of Krylov subspace methods for solving large sparse linear systems in various fields, including circuit simulations. MPK offers a potential for matrix reuse in cache, which can accelerate memory-bound sparse solvers. Additionally, many sparse matrices encountered in applications are symmetric, allowing us to reduce the memory footprint for SpMVs by half. However, reusing the matrix introduces data dependencies between subsequent SpMVs, and symmetric SpMVs can result in data conflicts during shared-memory parallelization. Previous research has often focused on either matrix reuse or symmetry, failing to leverage both aspects effectively. This paper proposes a unified, memory-efficient approach called Me-MPK that takes advantage of both cache reuse and matrix symmetry for MPK on shared-memory multi-core systems. We first introduce a unified dependency graph for a sparse matrix, which represents all potential data dependencies and conflicts. Next, we perform architecture-aware recursive partitioning on this graph to create subgraphs and formulate a separating subgraph that decouples all dependencies and conflicts among the subgraphs. These independent subgraphs are then scheduled for parallel execution of SpMV or symmetric SpMV in a specified order to optimize cache reuse. We apply Me-MPK in two s-Step Krylov subspace solvers, and our evaluations show that Me-MPK significantly outperforms the current state-of-the-art solutions, delivering an average speedup of up to 2.00X and 1.86X on X86 and ARM CPUs, respectively. As a result, we achieve overall speedup in the sparse solvers of up to $\mathbf{1. 6 5 X}$ and $\mathbf{1. 5 8 X}$.

Sparse iterative solvers are commonly used in various fields. However, certain essential kernels of these solvers, such as sparse triangular solves (SpTRSV), present significant challenges for efficient parallelization due to data dependencies . Previous methods, like level-scheduling or multi-coloring, typically involve creating a Task Dependency Graph (TDG) to represent data dependencies and identify independent sets from the TDG for parallel execution. However, these approaches often result in limited parallelism with substantial synchronization overheads or negatively impact the solver convergence rate. This article introduces DCSolver , a Divide-and-Conquer (DC) framework designed to efficiently parallelize sparse solvers with data dependencies on GPUs. To achieve this, we break down the solver TDG into independent subgraphs, allowing us to exploit both coarse-grained and fine-grained parallelism. To efficiently allocate GPU threads for subgraphs with varying degrees of parallelism, we have developed an adaptive in-warp scheduling strategy. Additionally, we propose a hybrid parallelization scheme in DCSolver, which involves employing different parallel approaches for different DC recursions to achieve a more optimal balance between parallelism and convergence for solvers. To evaluate the effectiveness of DCSolver, we apply it to two preconditioned Krylov subspace solvers and an unstructured mesh Computational Fluid Dynamics (CFD) solver. Our results show that when compared with the state-of-the-art methods, DCSolver accelerates the time-to-solution of solvers by an average speedup of up to 26.19X.