Spectral analysis of saddle-point matrices from optimization problems with elliptic PDE constraints

The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. They uncover the existence of an hidden structure in these matrix sequences, namely, they show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. They show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that they deal with saddle--point matrices. Finally, they exploit it to propose an optimal preconditioner strategy for the GMRES, and Flexible--GMRES methods.

Spectral Analysis, Properties and Nonsingular Preconditioners for Singular Saddle Point Problems

We first derive some explicit bounds on the spectra of generalized non-symmetric singular or nonsingular saddle point matrices. Then we propose two new nonsingular preconditioners for solving generalized singular saddle point problems, and show that GMRES determines a solution without breakdown when applied to the resulting preconditioned systems with any initial guess. Furthermore, the detailed spectral properties of the preconditioned systems are analyzed. The nonsingular preconditioners are also applied to solve the singular finite element saddle point systems arising from the discretization of the Stokes problems to test their performance.