Solving finite difference linear systems on GPUs: CUDA based Parallel Explicit Preconditioned Biconjugate Conjugate Gradient type Methods

Abstract

During the last decades, explicit approximate inverse preconditioning methods have been used for efficiently solving sparse linear systems on multiprocessor systems. The effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel. A new parallel computational technique is proposed for the parallelization of the explicit preconditioned conjugate gradient type method on a Graphics Processing Unit (GPU). The proposed parallel methods have been implemented using Compute Unified Device Architecture (CUDA) developed by NVIDIA. The inherently parallel operations between vectors and matrices involved in the explicit preconditioned biconjugate conjugate gradient type schemes exhibit significant amounts of loop-level parallelism because of the matrix–vector and the vector–vector products that can lead to high performance gain on the GPU systems, specifically designed for such computations. Finally, numerical results for the performance of the explicit preconditioned biconjugate conjugate gradient type method for solving characteristic two dimensional boundary value problems, using the finite difference method, on a massive multiprocessor interface on a GPU are presented. The CUDA implementation issues of the proposed method are also discussed.