ABSTRACT
This chapter focuses on several approaches to constructing the optimal and nearly optimal preconditioners mentioned in § 0.3. It is noteworthy that in all of these constructions it observes interplay between such basic notions as splitting, partitioning, block elimination, factorization, inner iterations, and spectral equivalence. These notions probably form the primary means of dealing with complicated problems, most particularly with problems involving domains of complicated geometry. The chapter emphasizes that it is impossible to give a universal prescription for choosing the most effective method for solving concrete problems, but a basic understanding of the nature of these methods is essential for making effective choices for specific applications. A wide class of model factored operators can be constructed using various generalizations of incomplete factorization. A very interesting use of factored matrices was given in connection with the use of hierarchical bases and splitting considered in § 2.5.
