TABLE OF CONTENTS This chapter covers a few alternative methods for preconditioning a linear system. These methods are suitable when the desired goal is to maximize parallelism. The simplest approach is the diagonal (or Jacobi) preconditioning. Often, this preconditioner is not very useful, since the number of iterations of the resulting iteration tends to be much larger than the more standard variants, such as incomplete LU (ILU) or symmetric successive overrelaxation (SSOR). When developing parallel preconditioners, one should be aware that the benefits of increased parallelism are not outweighed by the increased number of computations. The main question to ask is whether or not it is possible to find preconditioning techniques that have a high degree of parallelism, as well as good intrinsic qualities.
As seen in the previous chapter, a limited amount of parallelism can be extracted from the standard preconditioners, such as ILU and SSOR. Fortunately, a number of alternative techniques can be developed that are specifically targeted at parallel environments. These are preconditioning techniques that would normally not be used on a standard machine, but only for parallel computers. There are at least three such types of techniques discussed in this chapter. The simplest approach is to use a Jacobi or, even better, a block Jacobi approach. In the simplest case, a Jacobi preconditioner may consist of the diagonal or a block diagonal of A. To enhance performance, these preconditioners can themselves be accelerated by polynomial iterations, i.e., a second level of preconditioning called polynomial preconditioning.
