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Efficient Preconditioning Strategies for the Multilevel Fast Multipole Algorithm

For the iterative solutions of the integral equation methods employing the multilevel fast multipole algorithm (MLFMA), effective preconditioning techniques should be developed for robustness and efficiency. Preconditioning techniques for such problems can be broadly classified as fixed preconditioners that are generated from the sparse near-field matrix and variable ones that can make use of MLFMA with the help of the flexible solvers. Among fixed preconditioners, we show that an incomplete LU preconditioner depending on threshold (ILUT) is very successful in sequential implementations, provided that pivoting is applied whenever the incomplete factors become unstable. For parallel preconditioners, sparse approximate inverses (SAI) can be used; however, they are not as successful as ILUT for the electric-field integral equation. For a remedy, we employ variable preconditioning, and we iteratively solve the near-field system in each major iteration. However, for very large systems, neither of these methods succeeds to reduce the iteration counts as desired because of the thinning of the near-field matrices for increasing problem sizes. Considering this fact, we develop a preconditioner using MLFMA, with which we solve an approximate system. Respective advantages of these different preconditioners are demonstrated on a variety of problems ranging in both geometry and size.