Using GPU-Based Computing to Solve Large Sparse Systems of Linear Equations

Often thought of as tools for image rendering or data visualization, graphics processing units (GPU) are becoming increasingly popular in the areas of scientific computing due to their low cost massively parallel architecture. With the introduction of CUDA C by NVIDIA and CUDA enabled GPUs, the ability to perform general purpose computations without the need to utilize shading languages is now possible. One such application that benefits from the capabilities provided by NVIDIA hardware is computational continuum mechanics (CCM). The need to solve sparse linear systems of equations is common in CCM when partial differential equations are discretized. Often these systems are solved iteratively using domain decomposition among distributed processors working in parallel. In this paper we explore the benefits of using GPUs to improve the performance of sparse matrix operations, more specifically, sparse matrix-vector multiplication. Our approach does not require domain decomposition, so it is simpler than corresponding implementation for distributed memory parallel computers. We demonstrate that for matrices produced from finite element discretizations on unstructured meshes, the performance of the matrix-vector multiplication operation is just under 13 times faster than when run serially on an Intel i5 system. Furthermore, we show that when used in conjunction with the biconjugate gradient stabilized method (BiCGSTAB), a gradient based iterative linear solver, the method is over 13 times faster than the serially executed C equivalent. And lastly, we emphasize the application of such method for solving Poisson’s equation using the Galerkin finite element method, and demonstrate over 10.5 times higher performance on the GPU when compared with the Intel i5 system.