Parallel LU Decomposition for Solving Dense Linear Systems

快速求解线性方程组是电磁场、流体力学、结构力学等许多工程问题的核心，LU分解是求解稠密线性方程组最常用的方法之一。该文详细描述了LU分解的原理和过程，基于此过程的特点，在CUDA (Compute Unified Device Architecture)并行环境下，提出一种的并行计算方法，该方法可以将计算任务划分为若干小问题进行求解,各部分求解完成后再通过前向回代和后向回代，即可获得最终解。最后通过实验证明，该方法能够充分利用GPU，有效地减少了数据间的通信开销，从而加快了求解速度，提高了计算效率。Rapid solving of linear systems is central to engineering challenges in electromagnetics, fluid dynamics, structural mechanics, and more. LU decomposition remains one of the most widely used methods for dense linear systems. This paper elaborates on the principles and process of LU decomposition. Leveraging its algorithmic characteristics, we propose a parallel computation framework on CUDA that decomposes the computational task into smaller sub-problems. After solving these sub-tasks independently, the final solution is reconstructed through forward and backward substitutions. Experimental results demonstrate that this method maximizes GPU utilization, significantly reduces inter-node communication overhead, and achieves accelerated solving speeds with improved computational efficiency.