A Matrix-free Preconditioner for the Helmholtz Equation based on the Fast Multipole Method

Fast multipole methods (FMM) were originally developed for accelerating $N$-body problems for particle-based methods. FMM is more than an $N$-body solver, however. Recent efforts to view the FMM as an elliptic Partial Differential Equation (PDE) solver have opened the possibility to use it as a preconditioner for a broader range of applications. FMM can solve Helmholtz problems with optimal $\mathcal{O}(N \log N)$ complexity, has compute-bound inner kernels, and highly asynchronous communication patterns. The combination of these features makes FMM an interesting candidate as a preconditioner for sparse solvers on architectures of the future. The use of FMM as a preconditioner allows us to use lower order multipole expansions than would be required as a solver because individual solves need not be accurate. This reduces the amount of computation and communication significantly and makes the time-to-solution competitive with state-of-the-art preconditioners. Furthermore, the high asynchronicity of FMM allows it to scale to much larger core counts than factorization-based and multilevel methods. We describe our tests in reproducible details with freely available codes.