Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems

Abstract

The importance of Schur complement based preconditioners are well-established for classical saddle point problems in R N × R M . In this paper we extend these results to multiple saddle point problems in Hilbert spaces X 1 × X 2 × · · · × X n . For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems.

Preconditioners for Krylov subspace methods: An overview

“…For the marker-and-cell finite difference scheme, life is even easier since we can replace S by the identity matrix. We note that the preconditioner (8) has been generalized to certain problems containing more blocks [164][165][166][167], and that in general block diagonal preconditioning is a popular choice for symmetric saddle point problems [168][169][170][171]. The operator preconditioning approach can also be more widely employed; for more examples, see [55].…”

“…where Q denotes a mass matrix and K a stiffness matrix, and with x 1 , x 2 corresponding to the discretizations of u, f . Poisson control problems of this type have been considered from both the infinite-dimensional [52,55,167,174,336] and finite-dimensional [281,335,337,338] points of view. In particular by viewing the problem on the infinite-dimensional level, for instance by considering nonstandard norms, and then discretizing the resulting infinite-dimensional interpretation of the preconditioner, one may arrive at the optimal preconditioner [52,55]:…”

“…Poisson control problems of this type have been considered from both the infinite‐dimensional [52,55,167,174,336] and finite‐dimensional [281,335,337,338] points of view. In particular by viewing the problem on the infinite‐dimensional level, for instance by considering nonstandard norms, and then discretizing the resulting infinite‐dimensional interpretation of the preconditioner, one may arrive at the optimal preconditioner [52,55]: This methodology may also be applied in the setting of constraint preconditioners [336], for instance.…”

An Augmented Lagrangian Uzawa Iterative Method for Solving Double Saddle-Point Systems with Semidefinite (2,2) Block and Its Application to DLM/FD Method for Elliptic Interface Problems

“…3) itself also owns a saddle-point structure, the linear algebraic system (1.1) is thus treated as a class of double saddle-point system, and fits the definition of a multiple saddle-point operator as given in [21]. Recently, there have been several literatures on the iterative method for solving such three-by-three block systems where the double saddle-point structure, instead of the single saddle-point structure, is studied and used for the construction of iterative method in order to reduce the overall workload of the iteration.…”

Robust multigrid solvers for the biharmonic problem in isogeometric analysis

“…[4]), or Schur complement preconditioners (cf. [5,6]). For the latter, also the three dimensional version of the biharmonic problem is of interest.…”