We consider conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle-point problems or, in particular, regularizations thereof. Such methods require preconditioners that preserve certain sub-blocks from the original systems but allow considerable flexibility for the remaining blocks. We construct a number of families of implicit factorizations that are capable of reproducing the required sub-blocks and (some) of the remainder. These generalize known implicit factorizations for the unregularized case. Improved eigenvalue clustering is possible if additionally some of the noncrucial blocks are reproduced. Numerical experiments confirm that these implicit-factorization preconditioners can be very effective in practice.
Dollar, H. S., Gould, N. I. M., Schilders, W. H. A., & Wathen, A. J. (2006). Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems. SIAM Journal on Matrix Analysis and Applications, 28(1), 170-189. https://doi.org/10.1137/05063427X