GPU Accelerated Matrix Solution Using Novel Preconditioner for Three Dimensional Laguerre-FDTD Method

Conventionally, the large sparse matrix equation (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula>) generated by the Laguerre-FDTD method is computed using direct matrix solvers, which is often numerically expensive and computationally slow. In this work, we demonstrate an innovative approach to replace direct matrix solver with an iterative algorithm for the Laguerre-FDTD method. A novel preconditioner, specifically targeted to improve the convergence rate of biconjugate gradient stabilized solver (BiCGSTAB), is derived and implemented in the Laguerre-FDTD method. Compared with the classical Jacobi preconditioner, the proposed preconditioner achieves on average an improvement of more than 1.3× in the convergence rate. To further leverage the computational efficiency, a modified sparse matrix- vector multiplication algorithm is proposed and implemented using a General-Purpose Graphics Processing Unit (GPGPU). The new algorithm ensures that all computations are performed within the GPU, with minimum number of device-to-host data transfer and global memory access. With GPU's accelerated computing capability, the proposed solver achieves more than 5× computational speed up with respect to a high performance CPU-based direct solver on average. In addition, due to the intrinsic memory efficient nature of iterative solver, our approach also shows maximally more than 31× reduction in memory consumption against the direct solver. Various numerical examples are simulated to validate the capability and improvement of the proposed method.