Optimal Preconditioners for Hybrid Direct-Iterative $\mathcal {H}$-Matrix Solvers in Boundary Element Methods

The paper proposes a new approach to the fast solution of matrix equations resulting from boundary element discretization of integral equations. By hybridizing fast iterative $\mathcal {H}$-matrix solvers with a fast direct $\mathcal {H}$-matrix preconditioner factorization, we create a framework that can be tuned between the extremes of a direct solver and a unpreconditioned iterative solver. This tuning is largely achieved using a single numerical parameter representing the preconditioner tolerance. A more complicated scheme involving two different tolerances is also briefly considered. The proposed framework is demonstrated on a high-order accurate Locally Corrected Nyström solution of surface integral equations for PEC targets. Examples consider various scattering problems including those featuring strong physical resonances. We show that appropriately choosing the preconditioner tolerance achieves the prescribed solution accuracy with minimal CPU time. Expanding from one to two tolerance parameters further enhances the framework by providing the flexibility to dynamically adjust tolerance, enabling higher compression while maintaining accuracy and fast convergence. This adaptive strategy offers significant potential for optimizing the balance between memory usage and CPU time in the future.