SCAIM Seminar: Dominique Orban

A preconditioned variant of the Golub-Kahan bidiagonalization process recently proposed by Arioli and Orban allows us to establish that SYMMLQ and MINRES applied to least-squares problems in symmetric saddle-point form perform redundant work and are combinations of methods such as LSQR and LSMR. A well-chosen preconditioner allows us to formulate a projected variant of the Golub-Kahan process that forms the basis of specialized numerical methods for linear least-squares problems with linear equality constraints. As before, full-space methods such as SYMMLQ and MINRES applied to the symmetric saddle-point system defining the optimality conditions of such problems perform redundant work and are combinations of projected variants of LSQR and LSMR. We establish connections between numerical methods for least-squares problems, full-space methods and the projected and constraint-preconditioned Krylov methods of Gould, Orban and Rees.