A Cholesky preconditioned conjugate gradient algorithm on GPU for the 3D parabolic equation

Abstract

For solving the large sparse linear systems to which numerical solution of the three-dimensional (3D) parabolic equation leads, an efficient parallel preconditioned conjugate gradient algorithm with the modified incomplete Cholesky (MIC) preconditioner on the GPU (MICPCGA) is proposed. In our proposed method, for this case, we overcome the drawback that the MIC preconditioner is generally difficult to parallelise on the GPU owing to the forward/backward substitutions, and thus present an efficient parallel implementation method on the GPU (GPUFBS). In addition, the vector operations are optimised by grouping several vector operations into a single kernel, and an inner-product kernel is suggested, and a kernel for the sparse matrix-vector multiplication in the CUSPARSE library is adopted. Numerical results show that our proposed GPUFBS and MICPCGA both can achieve a significant speedup, and compared to an approximate inverse SSOR preconditioned conjugate gradient algorithm (SSORPCGA), our proposed MICPCGA not only obtains a bigger speedup, but also has higher precision in solving the 3D parabolic equation.