  In this article a new family of preconditioners is introduced for symmetric
positive definite linear systems. The new preconditioners, called the AWG
preconditioners (for Algebraic-Woodbury-GenEO) are constructed algebraically.
By this, we mean that only the knowledge of the matrix A for which the linear
system is being solved is required. Thanks to the GenEO spectral coarse space
technique, the condition number of the preconditioned operator is bounded
theoretically from above. This upper bound can be made smaller by enriching the
coarse space with more spectral modes.The novelty is that, unlike in previous
work on the GenEO coarse spaces, no knowledge of a partially non-assembled form
of A is required. Indeed, the spectral coarse space technique is not applied
directly to A but to a low-rank modification of A of which a suitable
non-assembled form is known by construction. The extra cost is a second (and to
this day rather expensive) coarse solve in the preconditioner. One of the AWG
preconditioners has already been presented in the short preprint [38]. This
article is the first full presentation of the larger family of AWG
preconditioners. It includes proofs of the spectral bounds as well as numerical
illustrations.
