Fast Iterative Solution of Multiple Right-Hand Sides MoM Linear Systems on CPUs and GPUs Computers

The method of moments discretization of boundary integral equations typically leads to dense linear systems. When a single operator is used for multiple excitations, these systems have the same coefficient matrix and multiple right-hand sides (RHSs). Direct solution methods are computationally attractive to use for the solution because they require one-round forward/backward substitution for each RHS, whereas the iterative solution necessitates a full-round iteration for each separate vector. However, direct methods become impractical to use even on modern parallel computers when the dimension $n$ is large, due to their ${\mathcal{O}}(n^{2})$ memory and ${\mathcal{O}}(n^{3})$ computational complexity. In this article, we present experiments with a block iterative Krylov subspace method that solves the entire set of systems at once. The proposed method performs a block size reduction during the construction of the Krylov subspace and combines spectral preconditioning and eigenvalue recycling techniques to mitigate memory costs and reduce the number of iterations of conventional block Krylov solvers. Our experiments demonstrate the potential of our method for efficiently solving linear systems with many RHSs arising from integral equation-based engineering applications fast and efficiently on both CPUs and graphics processing units (GPUs).