Polynomial Preconditioning of Power System Matrices with Graphics Processing Units

Programmable graphics processing units (GPUs) currently offer the best ratio of floating point computational throughput to price for commodity processors, outdistancing same-generation CPUs by an order of magnitude, which has in turn led to their widespread adoption in a variety of computationally demanding fields. Adapting power system simulations to these processors is complicated by the unique hardware architecture of GPUs, which precludes the usage of direct linear system solvers based on Gaussian elimination. Krylov subspace methods are better suited to the GPU architecture, yet the ill-conditioned nature of power system matrices requires substantial preconditioning to ensure robustness of these methods. To reduce the time spent on preconditioning, we have developed a GPU-based preconditioner designed specifically to handle the large, sparse matrices typically encountered in power system simulations. The preconditioning technique used, based on Chebyshev polynomials, is described in detail, as are the design decisions made in adapting this algorithm to the GPU. Evaluation of the performance of the GPU-based preconditioner on a variety of sparse matrices, ranging in size from 30 x 30 to 3948 x 3948, shows significant computational savings relative to a CPU-based implementation of the same preconditioner and a typical incomplete LU (ILU) preconditioner.