GPU-accelerated preconditioned GMRES method for two-dimensional Maxwell's equations

In this study, for two-dimensional Maxwell's equations, an efficient preconditioned generalized minimum residual method on the graphics processing unit GPUPGMRES is proposed to obtain numerical solutions of the equations that are discretized by a multisymplectic Preissmann scheme. In our proposed GPUPGMRES, a novel sparse matrix–vector multiplication SpMV kernel is suggested while keeping the compressed sparse row CSR intact. The proposed kernel dynamically assigns different number of rows to each thread block, and accesses the CSR arrays in a fully coalesced manner. This greatly alleviates the bottleneck of many existing CSR-based algorithms. Furthermore, the vector-operation and inner-product decision trees are automatically constructed. These kernels and their corresponding optimized compute unified device architecture parameter values can be automatically selected from the decision trees for vectors of any size. In addition, using the sparse approximate inverse technique, the preconditioner equation solving falls within the scope of SpMV. Numerical results show that our proposed kernels have high parallelism. GPUPGMRES outperforms a recently proposed preconditioned GMRES method, and a preconditioned GMRES implementation in the AmgX library. Moreover, GPUPGMRES is efficient in solving the two-dimensional Maxwell's equations.