This paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices A. By resorting to rational functions of A, the algorithm decomposes the spectrum of A into two disjoint regions and approximates the restriction of A1 on these regions separately. We show a systematic way to construct these rational functions so that they can be applied stably and inexpensively. An attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism. Moreover, the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form AI, which can be useful in some applications. The efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems, including problems arising from the Helmholtz equation in three dimensions.
Bibliographical noteFunding Information:
This work was supported by NSF grants DMS-1216366 and DMS-1521573 and by the Minnesota Supercomputing Institute.
© 2017 Society for Industrial and Applied Mathematics.
- Approximate inverse
- Cauchy integral
- De ation
- Helmholtz equation
- Incomplete LU
- Rational function