Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. I: Theory. (English) Zbl 1078.65027

de Sturler, Eric; Liesen, Jörg

SIAM J. Sci. Comput. 26, No. 5, 1598-1619 (2005).

Block-diagonal preconditioners and the resulting preconditioned systems are introduced. Then the results of a study on the properties of the preconditioned matrices, in particular their eigendecompositions, are presented. The fixed point iteration and its related system are derived. The spectral radius of the fixed point iteration matrix and the spectrum of the related system matrix are also analyzed. Why it is more efficient to apply the GREMS algorithm to the related system than to the preconditioned system is explained. The paper concludes with results of numerical experiments and a discussion of scaling of nonlinear constraints in this constrained optimization problem to improve the convergence in each Newton step.

Reviewer: R. P. Tewarson (Stony Brook)

MSC:

65F10	Iterative numerical methods for linear systems
65F35	Numerical computation of matrix norms, conditioning, scaling
65K05	Numerical mathematical programming methods
90C30	Nonlinear programming

Keywords:

saddle point systems; indefinite systems; eigenvalue bounds; fixed point iteration; GREMS algorithm; numerical experiments; scaling; convergence; Krylov subspace methods; preconditioning; constrained optimization; mesh-flattening