Sequential and parallel preconditioners for large-scale integral-equation problems

Abstract

For efficient solutions of integral-equation methods via the multilevel fast multipole algorithm (MLFMA), effective preconditioners are required. In this paper we review appropriate preconditioners that have been used for sparse systems and developed specially in the context of MLFMA. First we review the ILU-type preconditioners that are suitable for sequential implementations. We can make these preconditioners robust and efficient for integral-equation methods by making appropriate selections and by employing pivoting to suppress the instability problem. For parallel implementations, the sparse approximate inverse or the iterative solution of the near-field system enables fast convergence up to certain problem sizes. However, for very large problems, the near-field matrix itself becomes insufficient to approximate the dense system matrix and preconditioners generated from the near-field interactions cannot be effective. Therefore, we propose an approximation strategy to MLFMA to be used as an effective preconditioner. Our numerical experiments reveal that this scheme significantly outperforms other preconditioners. With the combined effort of effective preconditioners and an efficiently parallelized MLFMA, we are able to solve problems with tens of millions of unknowns in a few hours. We report the solution of integral-equation problems that are among the largest in their classes.

Preconditioning large integral-equation problems involving complex targets

“…The AMLFMA preconditioner, on the other hand, uses a cheap version of MLFMA for MVM, hence uses more than what is provided by the nearfield interactions. We have already reported the solution of a 22-million-unknown patch problem and verified its accuracy by comparing the MLFMA solution with a physical-optics solution [4]. In the next section, we will provide the solutions of more complex and real-life problems, and discuss the difficulties that arise when the sizes of matrix equations reach millions of unknowns.…”

“…We have previously reported three effective preconditioning techniques, i.e., a sparse approximate inverse (SAI) preconditioner, the iterative near-field (INF) preconditioner, and the approximate MLFMA (AMLFMA) preconditioner [4]. The SAI preconditioner is a low-cost, but effective preconditioner generated from the near-field matrix.…”

Extended near field preconditioner for the analysis of large problems using the nested‐FMM‐FFT algorithm

“…However, this assumption usually fails for large-scale problems. Looking for a better treatment of this kind of problems, in [15] SAI is used for the iterative solution of the near-field system with a inner-outer scheme and then the original system is preconditioned by this near-field solution.…”

“…Therefore, taking into account the limitations imposed by the FMM based algorithms, several extended preconditioners have been implemented to increase the convergence rates without increasing significantly the computational complexity [10][11][12][13][14][15].…”

Experiments With Lanczos Biconjugate a-Orthonormalization Methods for Mom Discretizations of Maxwell's Equations

“…The GMRES method [43] (or sometimes its flexible variant FGMRES [41]) is broadly used in application codes due to its robustness and smooth convergence, see e.g., the experiments reported in [6,35] for solving very large boundary element equations involving million discretization points. On the other hand, iterative methods based on Lanczos biconjugation, such as BiCGSTAB [56] and QMR [20] can be cheaper in terms of memory requirements but may converge more slowly especially on realistic geometries [6,36].…”