Analysis and Design Techniques towards High-Performance and Energy-Efficient Dense Linear Solvers on GPUs

Graphics Processing Units (GPUs) are widely used in accelerating dense linear solvers. The matrix factorizations, which dominate the runtime for these solvers, are often designed using a hybrid scheme, where GPUs perform trailing matrix updates, while the CPUs perform the panel factorizations. Consequently, hybrid solutions require high-end CPUs and optimized CPU software in order to deliver high performance. Furthermore, they lack the energy efficiency inherent for GPUs due to the use of less energy-efficient CPUs, as well as CPU-GPU communications. This paper presents analysis and design techniques that overcome the shortcomings of the hybrid algorithms, and allow the design of high-performance and energy-efficient dense LU and Cholesky factorizations that use GPUs only. The full GPU solution eliminates the need for a high-end CPU and optimized CPU software, which leads to a better energy efficiency. We discuss different design choices, and introduce optimized GPU kernels for panel factorizations. The developed solutions achieve
$90+$
percent of the performance of optimized hybrid solutions, while improving the energy efficiency by 50 percent. They outperform the vendor library by 30-50 percent in single precision, and 15-50 percent in double precision. We also show that hybrid designs trail the proposed solutions in performance when optimized CPU software is not available.