On A Class of Limited Memory Preconditioners For Large Scale Linear Systems With Multiple Right-Hand Sides

This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectralLMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see [8] and [20], respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.