Sparse matrix-vector multiplication (SpMV) is a crucial computing kernel with widespread applications in iterative algorithms. Over the past decades, research on SpMV optimization has made remarkable strides, giving rise to various optimization contributions. However, the comprehensive and systematic survey that introduces, analyzes, discusses, and summarizes the SpMV work in recent years is currently lacking. Aiming to fill this gap, this paper compares existing techniques and analyzes their strengths and weaknesses. We conduct an in-depth overview of the important techniques that optimize SpMV on modern architectures, which we specifically classify as classic, auto-tuning, machine learning, and mixed-precision-based optimization. We also elaborate on the hardware-based architectures, including CPU, GPU, and FPGA. We present an experimental evaluation that compares the performance of state-of-the-art SpMV implementations. Based on our findings, we identify several challenges and point out future research directions. This survey intends to provide researchers with a comprehensive understanding of SpMV optimization on modern architectures and guide future work.

• Citation
• Papers
  • SpMV Related Surveys
  • Sparse Compression Formats
    • Basic Compression Formats
    • New Compression Formats
      • Regular Slicing
      • Regular Blocking
      • Irregular Compressing
      • Bit or Byte Compressing
      • Hybrid Encoding
      • Other Variants
  • Auto-Tuning Based Algorithm
    • Offline Auto-Tuning
    • Online Auto-Tuning
  • Machine-Learning Based Algorithm
    • Format or Algorithm Selection
    • Parameter Prediction
    • Performance Prediction
  • Mixed Precision Based Optimization
    • Mixed-Precision Iterative Solving Algorithms
    • Mixed-Precision SpMV
  • Architecture Oriented Optimization
    • CPU
    • GPU
      • Single GPU
      • Multiple GPUs
    • FPGA
    • Processing in Memory
    • Heterogeneous Platform
    • Distributed Platform
• Sparse Libraries

@misc{gao2024spmvsurvey,
     title={A Systematic Literature Survey of Sparse Matrix-Vector Multiplication},
     author={Jianhua Gao and Bingjie Liu and Weixing Ji and Hua Huang},
     year={2024},
     eprint={2404.06047},
     archivePrefix={arXiv},
     primaryClass={cs.DC},
     url={https://arxiv.org/abs/2404.06047},
}

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• Scalable Parallel Programming with CUDA: Is CUDA the parallel programming model that application developers have been waiting for?. John Nickolls, Ian Buck, Michael Garland, Kevin Skadron, ACM Queue (Queue), 2008 [DOI] [pdf]
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• Overcoming Load Imbalance for Irregular Sparse Matrices. Goran Flegar, Hartwig Anzt, The International Conference for High Performance Computing, Networking, Storage and Analysis (SC), 2017 [DOI]
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• AMF-CSR: Adaptive Multi-Row Folding of CSR for SpMV on GPU. Jianhua Gao, Weixing Ji, Jie Liu, Senhao Shao, Yizhuo Wang, Feng Shi, International Conference on Parallel and Distributed Systems (ICPADS), 2021 [DOI] [pdf]
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• Efficient Sparse Matrix-Vector Multiplication on CUDA. Nathan Bell, Michael Garland, NVIDIA Technical Report (NVR),2008 [DOI]
• Implementing Sparse Matrix-Vector Multiplication on Throughput-Oriented Processors. Nathan Bell, Michael Garland, Supercomputing Conference (SC), 2009 [DOI] [pdf]
• Efficient Sparse Matrix-Vector Multiplication on Cache-Based GPUs. István R eguly, Mike Giles, Innovative Parallel Computing (InPar), 2012 [DOI] [pdf]
• Automatic Tuning of Sparse Matrix-Vector Multiplication for CRS Format on GPUs. Hiroki Yoshizawa, Daisuke Takahashi, IEEE International Conference on Computational Science and Engineering (CSE), 2012 [DOI] [pdf]
• Revisiting Thread Configuration of SpMV Kernels on GPU: A Machine Learning Based Approach. Jianhua Gao, Weixing Ji, Jie Liu, Yizhuo Wang, Feng Shi, Journal of Parallel and Distributed Computing (JPDC), 2024 [DOI] [pdf]
• Taming Irregular EDA Applications on GPUs. Yangdong Deng, Bo David Wang, Shuai Mu, IEEE International Conference on Computer-Aided Design (ICCAD), 2009 [DOI] [pdf]
• Parallel GMRES Solver for Fast Analysis of Large Linear Dynamic Systems on GPU Platforms. Kai He, Sheldon X.-D. Tan, Hengyang Zhao, Xue-Xin Liu, Hai Wang, Guoyong Shi, Integration, 2016 [DOI] [pdf]
• Sparse Matrix Computations on Manycore GPU’s. Michael Garland, Design Automation Conference (DAC), 2008 [DOI] [pdf]
• VCSR: An Efficient GPU Memory-Aware Sparse Format. Elmira Karimi, Nicolas Bohm Agostini, Shi Dong, David Kaeli, IEEE Transactions on Parallel and Distributed Systems (TPDS), 2022 [DOI] [pdf]
• Optimization of Sparse Matrix-Vector Multiplication for CRS Format on NVIDIA Kepler Architecture GPUs. Daichi Mukunoki, Daisuke Takahashi, Computational Science and Its Applications (ICCSA), 2013 [DOI]
• DASP: Specific Dense Matrix Multiply-Accumulate Units Accelerated General Sparse Matrix-Vector Multiplication. Yuechen Lu, Weifeng Liu, Supercomputing Conference (SC), 2023 [DOI] [pdf]
• Model-Driven Autotuning of Sparse Matrix-Vector Multiply on GPUs. Jee Whan Choi, Amik Singh, Richard W. Vuduc, ACM SIGPLAN Notices (ACM SIGPLAN Not), 2010 [DOI] [pdf]
• LightSpMV: Faster CSR-Based Sparse Matrix-Vector Multiplication on CUDA-Enabled GPUs. Yongchao Liu, Bertil Schmidt, International Conference on Application Specific Systems, Architectures and Processors (ASAP), 2015 [DOI] [pdf]
• LightSpMV: Faster CUDA-Compatible Sparse Matrix-Vector Multiplication Using Compressed Sparse Rows. Yongchao Liu, Bertil Schmidt, Journal of Signal Processing Systems (JSPS), 2018 [DOI]
• TileSpMV: A Tiled Algorithm for Sparse Matrix-Vector Multiplication on GPUs. Yuyao Niu, Zhengyang Lu, Meichen Dong, Zhou Jin, Weifeng Liu, Guangming Tan, International Symposium on Parallel and Distributed Processing (IPDPS), 2021 [DOI] [pdf]
• Auto-Tuning CUDA Parameters for Sparse Matrix-Vector Multiplication on GPUs. Ping Guo, Liqiang Wang, International Conference on Computational and Information Sciences (ICCIS), 2010 [DOI] [pdf]
• Automatically Generating and Tuning GPU Code for Sparse Matrix-Vector Multiplication from a High-Level Representation. Dominik Grewe, Anton Lokhmotov, Proceedings of the Fourth Workshop on General Purpose Processing on Graphics Processing Units (GPGPU), 2011 [DOI] [pdf]

• Fast Conjugate Gradients with Multiple GPUs. Ali Cevahir, Akira Nukada, Satoshi Matsuoka, Computational Science (ICCS), 2009 [DOI] [pdf]
• Performance Optimization for SpMV on Multi-GPU Systems Using Threads and Multiple Streams. Ping Guo, Changjiang Zhang, International Symposium on Computer Architecture and High Performance Computing Workshops (SBAC-PADW), 2016 [DOI] [pdf]
• Multi-GPU Implementation of the Uniformization Method for Solving Markov Models. Marek Karwacki, Beata Bylina, Jarosław Bylina, Federated Conference on Computer Science and Information Systems (FedCSIS), 2012 [DOI] [pdf]
• Analysis and Performance Estimation of the Conjugate Gradient Method on Multiple GPUs. Mickeal Verschoor, Andrei C. Jalba, Parallel Computing (PC), 2012 [DOI] [pdf]
• Preconditioned GMRES Solver on Multiple-GPU Architecture. Bo Yang, Hui Liu, Zhangxin Chen, Computers & Mathematics with Applications (CMA), 2016 [DOI] [pdf]
• A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. George Karypis, Vipin Kumar, SIAM Journal on Scientific Computing (SJSC), 1998 [DOI] [pdf]
• Adaptive Optimization Modeling of Preconditioned Conjugate Gradient on Multi-GPUs. Jiaquan Gao, Yu Wang, Jun Wang, Ronghua Liang, ACM Transactions on Parallel Computing (TOPC), 2016 [DOI]
• A Multi-GPU Parallel Optimization Model for the Preconditioned Conjugate Gradient Algorithm. Jiaquan Gao, Yuanshen Zhou, Guixia He, Yifei Xia, Parallel Computing (PC), 2017 [DOI] [pdf]
• A Novel Multi-Graphics Processing Unit Parallel Optimization Framework for the Sparse Matrix-Vector Multiplication. Jiaquan Gao, Yu Wang, Jun Wang, Concurrency and Computation: Practice and Experience (CCPE), 2017 [DOI]
• P-cloth: Interactive Complex Cloth Simulation on Multi-GPU Systems Using Dynamic Matrix Assembly and Pipelined Implicit Integrators. Cheng Li, Min Tang, Ruofeng Tong, Ming Cai, Jieyi Zhao, Dinesh Manocha, ACM Transactions on Graphics (TOG), 2020 [DOI]
• MSREP: A Fast yet Light Sparse Matrix Framework for Multi-GPU Systems. Jieyang Chen, Chenhao Xie, Jesun S Firoz, Jiajia Li, Shuaiwen Leon Song, Kevin Barker, Mark Raugas, Ang Li, arXiv, 2022 [DOI] [pdf]
• Exploring the Multiple-GPU Design Space. Dana Schaa, David Kaeli, International Symposium on Parallel and Distributed Processing (IPDPS), 2009 [DOI] [pdf]
• High Performance Multi-GPU SpMV for Multi-component PDE-Based Applications. Ahmad Abdelfattah, Hatem Ltaief, David Keyes, Euro-Par: Parallel Processing (Euro-Par), 2015 [DOI] [pdf]

• FPGA and GPU Implementation of Large Scale SpMV. Yi Shan, Tianji Wu, Yu Wang, Bo Wang, Zilong Wang, Ningyi Xu, Symposium on Application Specific Processors (SASP), 2010 [DOI] [pdf]
• A Case Study of Streaming Storage Format for Sparse Matrices. Shweta Jain-Mendon, Ron Sass, International Conference on Reconfigurable Computing and FPGAs (ReConFig), 2012 [DOI] [pdf]
• Towards a Universal FPGA Matrix-Vector Multiplication Architecture. Srinidhi Kestur, John D. Davis, Eric S. Chung, Annual IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM), 2012 [DOI] [pdf]
• An Energy Efficient Column-Major Backend for FPGA SpMV Accelerators. Yaman Umuroglu, Magnus Jahre, IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD), 2014 [DOI] [pdf]
• A High Memory Bandwidth FPGA Accelerator for Sparse Matrix-Vector Multiplication. Jeremy Fowers, Kalin Ovtcharov, Karin Strauss, Eric S. Chung, Greg Stitt, Annual IEEE Symposium on Field-Programmable Custom Computing Machines (FCCM), 2014 [DOI] [pdf]
• Using Machine Learning to Estimate Utilization and Throughput for OpenCL-Based Matrix-Vector Multiplication (MVM). Jannatun Naher, Clay Gloster, Christopher C. Doss, Shrikant S. Jadhav, IEEE Annual Computing and Communication Workshop and Conference (CCWC), 2020 [DOI] [pdf]
• A Vector Caching Scheme for Streaming FPGA SpMV Accelerators. Yaman Umuroglu, Magnus Jahre, Applied Reconfigurable Computing (ARC), 2015 [DOI]
• Random Access Schemes for Efficient FPGA SpMV Acceleration. Yaman Umuroglu, Magnus Jahre, Microprocessors and Microsystems (MICPRO), 2016 [DOI] [pdf]
• Efficient SpMV Operation for Large and Highly Sparse Matrices using Scalable Multi-way Merge Parallelization. Fazle Sadi, Joe Sweeney, Tze Meng Low, James C. Hoe, Larry Pileggi, Franz Franchetti, Proceedings of the 52nd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO), 2019 [DOI]
• A Streaming Dataflow Engine for Sparse Matrix-Vector Multiplication Using High-Level Synthesis. Mohammad Hosseinabady, Jose Luis Nunez-Yanez, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (TCAD), 2020 [DOI] [pdf]
• An FPGA Cached Sparse Matrix Vector Product (SpMV) for Unstructured Computational Fluid Dynamics Simulations. Guillermo Oyarzun, Daniel Peyrolon, Carlos Alvarez, Xavier Martorell, arXiv, 2021 [DOI] [pdf]
• Scaling up HBM Efficiency of Top-K SpMV for Approximate Embedding Similarity on FPGAs. Alberto Parravicini, Luca Giuseppe Cellamare, Marco Siracusa, Marco D. Santambrogio, Design Automation Conference (DAC), 2021 [DOI] [pdf] [code]
• Towards High-Bandwidth-Utilization SpMV on FPGAs via Partial Vector Duplication. Bowen Liu, Dajiang Liu, Asia and South Pacific Design Automation Conference (ASP-DAC),2023 [DOI] [pdf]
• FPGA-Based HPC Accelerators: An Evaluation on Performance and Energy Efficiency. Tan Nguyen, Colin MacLean, Marco Siracusa, Douglas Doerfler, Nicholas J. Wright, Samuel Williams, Concurrency Computation: Practice and Experience (CCPE), 2022 [DOI]
• Optimizing the Performance of the Sparse Matrix–Vector Multiplication Kernel in FPGA Guided by the Roofline Model. * Federico Favaro, Ernesto Dufrechou, Juan P. Oliver, Pablo Ezzatti, Micromachines, 2023* [DOI]
• Using Machine Learning to Estimate Utilization and Throughput for OpenCL-Based Matrix-Vector Multiplication (MVM). Jannatun Naher, Clay Gloster, Christopher C. Doss, Shrikant S. Jadhav, IEEE Annual Computing and Communication Workshop and Conference (CCWC), 2020 [DOI] [pdf]
• Hardware Acceleration of SpMV Multiplier for Deep Learning. Mahadurkar Mahesh, Nalesh Sivanandan, Kala S, International Symposium on VLSI Design and Test (VDAT), 2021 [DOI] [pdf]

• SpaceA: Sparse Matrix Vector Multiplication on Processing-in-Memory Accelerator. Xinfeng Xie, Zheng Liang, Peng Gu, Abanti Basak, Lei Deng, Ling Liang, IEEE Symposium on High-Performance Computer Architecture (HPCA), 2021 [DOI] [pdf]
• ABC-DIMM: Alleviating the Bottleneck of Communication in DIMM-based Near-Memory Processing with Inter-DIMM Broadcast. Weiyi Sun, Zhaoshi Li, Shouyi Yin, Shaojun Wei, Leibo Liu, Annual International Symposium on Computer Architecture (ISCA), 2021 [DOI] [pdf]
• SparseP: Towards Efficient Sparse Matrix Vector Multiplication on Real Processing-In-Memory Architectures. Christina Giannoula, Ivan Fernandez, Juan Gómez Luna, Nectarios Koziris, Georgios Goumas, Onur Mutlu, Proceedings of the ACM on Measurement and Analysis of Computing Systems (POMACS), 2022 [DOI] [code]

• Architecture- and Workload- Aware Heterogeneous Algorithms for Sparse Matrix Vector Multiplication. Sivaramakrishna Bharadwaj Indarapu, Manoj Maramreddy, Kishore Kothapalli, Proceedings of the 7th ACM India Computing Conference (COMPUTE), 2014 [DOI]
• Heterogeneous Sparse Matrix Computations on Hybrid GPU/CPU Platforms. Valeria Cardellini, Alessandro Fanfarillo, Salvatore Filippone, International Conference on Parallel Computing (ICPC), 2013 [DOI] [pdf]
• Performance Optimization Using Partitioned SpMV on GPUs and Multicore CPUs. Wangdong Yang, Kenli Li, Zeyao Mo, Keqin Li, IEEE Transactions on Computers (ITC), 2015 [DOI] [pdf]
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• Sparse Matrix Partitioning for Optimizing SpMV on CPU-GPU Heterogeneous Platforms. Akrem Benatia, Weixing Ji, Yizhuo Wang, Feng Shi, The International Journal of High Performance Computing Applications (IJHPCA), 2019 [DOI]
• A Comparison of Eleven Static Heuristics for Mapping a Class of Independent Tasks onto Heterogeneous Distributed Computing Systems. Tracy D. Braun, Howard Jay Siegel, Noah Beck, Ladislau L. Bölöni, Muthucumaru Maheswaran, Albert I. Reuther, James P. Robertson, Mitchell D. Theys, Bin Yao, Debra Hensgen, Richard F. Freund, Journal of Parallel and Distributed Computing (JPDC), 2001 [DOI] [pdf]
• Speculative Segmented Sum for Sparse Matrix-vector Multiplication on Heterogeneous Processors. Weifeng Liu, Brian Vinter, Parallel Computing (PC), 2015 [DOI] [pdf]
• Efficient Sparse Matrix-Vector Multiplication on x86-Based Many-Core Processors. Xing Liu, Mikhail Smelyanskiy, Edmond Chow, Pradeep Dubey, Proceedings of the 27th international ACM conference on International conference on supercomputing (ICS), 2013 [DOI [pdf]
• An Efficient SIMD Compression Format for Sparse Matrix-Vector Multiplication. Xinhai Chen, Peizhen Xie, Lihua Chi, Jie Liu, Chunye Gong, Concurrency and Computation: Practice and Experience (CCPE), 2018 [DOI]
• Optimizing and Auto-Tuning Scale-Free Sparse Matrix-Vector Multiplication on Intel Xeon Phi. Wai Teng Tang, Ruizhe Zhao, Mian Lu, Yun Liang, Huynh Phung Huyng, Xibai Li, International Symposium on Code Generation and Optimization (CGO), 2015 [DOI] [pdf]
• CVR: Efficient Vectorization of SpMV on x86 Processors. Biwei Xie, Jianfeng Zhan, Xu Liu, Wanling Gao, Zhen Jia, Xiwen He, Lixin Zhang, Proceedings of the 2018 International Symposium on Code Generation and Optimization (CGO), 2018 [DOI] [pdf] [code]
• VBSF: a New Storage Format for SIMD Sparse Matrix–Vector Multiplication on Modern Processors. Yishui Li, Peizhen Xie, Xinhai Chen, Jie Liu, Bo Yang, Shengguo Li, Chunye Gong, Xinbiao Gan, Han Xu, The Journal of Supercomputing (JS), 2020 [DOI]
• Towards Efficient SpMV on Sunway Manycore Architectures. Changxi Liu, Biwei Xie, Xin Liu, Wei Xue, Hailong Yang, Xu Liu, Proceedings of the 2018 International Conference on Supercomputing (ICS), 2018 [DOI]
• hpSpMV: A Heterogeneous Parallel Computing Scheme for SpMV on the Sunway TaihuLight Supercomputer. Yuedan Chen, Guoqing Xiao, Zheng Xiao, Wangdong Yang, IEEE International Conference on High Performance Computing and Communications (HPCC), 2019 [DOI] [pdf]
• tpSpMV: A Two-Phase Large-Scale Sparse Matrix-Vector Multiplication Kernel for Manycore Architectures. Yuedan Chen, Guoqing Xiao, Fan Wu, Zhuo Tang, Keqin Li, Information Sciences (IS), 2020 [DOI] [pdf]
• ahSpMV: An Autotuning Hybrid Computing Scheme for SpMV on the Sunway Architecture. Guoqing Xiao, Yuedan Chen, Chubo Liu, Xu Zhou, IEEE Internet of Things Journal (IOT), 2020 [DOI] [pdf]
• CASpMV: A Customized and Accelerative SpMV Framework for the Sunway TaihuLight. Guoqing Xiao, Kenli Li, Yuedan Chen, Wangquan He, Albert Y. Zomaya, Tao Li, IEEE Transactions on Parallel and Distributed Systems (TPDS), 2021 [DOI] [pdf]
• HASpMV: Heterogeneity-Aware Sparse Matrix-Vector Multiplication on Modern Asymmetric Multicore Processors. Wenxuan Li, Helin Cheng, Zhengyang Lu, Yuechen Lu, Weifeng Liu, IEEE International Conference on Cluster Computing (ISCA), 2023 [DOI] [pdf]

• SMVP Distribution Using Hypergraph Model and S-GBNZ Algorithm. Ichrak Mehrez, Olfa Hamdi-Larbi, International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC), 2013 [DOI] [pdf]
• Balancing Computation and Communication in Distributed Sparse Matrix-Vector Multiplication. Hongli Mi, Xiangrui Yu, Xiaosong Yu, Shuangyuan Wu, Weifeng Liu, IEEE/ACM International Symposium on Cluster Computing and the Grid (CCGRID), 2023 [DOI] [pdf]
• Scalability of Hybrid SpMV with Hypergraph Partitioning and Vertex Delegation for Communication Avoidance. Brian A. Page, Peter M. Kogge, International Conference on High Performance Computing & Simulation (HPCS), 2020 [DOI] [pdf]
• HYPE: Massive Hypergraph Partitioning with Neighborhood Expansion. Christian Mayer, Ruben Mayer, Sukanya Bhowmik, Lukas Epple, Kurt Rothermel, IEEE International Conference on Big Data (BigData), 2018 [DOI] [pdf] [code]
• A Jacobi_PCG Solver for Sparse Linear Systems on Multi-GPU Cluster. Shaozhong Lin, Zhiqiang Xie, The Journal of Supercomputing (JS), 2017 [DOI]
• Parallel Sparse Linear Solver with GMRES Method Using Minimization Techniques of Communications for GPU Clusters. Lilia Ziane Khodja, Raphaël Couturier, Arnaud Giersch & Jacques M. Bahi, The Journal of Supercomputing (JS), 2014 [DOI]
• Performance Analysis of Multicore and Multinodal Implementation of SpMV Operation. Beata Bylina, Jarosław Bylina, Przemysław Stpiczyński, Dominik Szałkowski, Federated Conference on Computer Science and Information Systems (FedCSIS), 2014 [DOI] [pdf]
• Adaptive Runtime Tuning of Parallel Sparse Matrix-Vector Multiplication on Distributed Memory Systems. Seyong Lee, Rudolf Eigenmann, Proceedings of the 22nd annual international conference on Supercomputing (ICS), 2008 [DOI] [pdf]
• ZAKI: A Smart Method and Tool for Automatic Performance Optimization of Parallel SpMV Computations on Distributed Memory Machines. Sardar Usman, Rashid Mehmood, Iyad Katib, Aiiad Albeshri, Saleh M. Altowaijri, Mobile Networks and Applications (MONET), 2019 [DOI]
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• Developing a Multi-GPU-Enabled Preconditioned GMRES with Inexact Triangular Solves for Block Sparse Matrices. Wenpeng Ma, Yiwen Hu, Wu Yuan, Xiazhen Liu, Mathematical Problems in Engineering (MPE), 2021 [DOI] [pdf]

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