Analysis on inexact block diagonal preconditioners for elliptic PDE-constrained optimization problems
The authors consider an efficient preconditioning technique to a linear elliptic distributed control problem in two-dimensional spaces. By following the discretize-then-optimize approach, a Galerkin finite element method is used to transform the control problem into a linear system of saddle point form. A class of inexact block diagonal preconditioners is introduced and the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices are estimated. Based on the Cholesky decomposition of the known matrices, a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system is also analyzed. The real and complex eigenvalues of the reduced linear system are discussed. It is shown that these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system. Some numerical experiments are presented to demonstrate the effectiveness and robustness of the two new preconditioners.
scientific article; zbMATH DE number 6929639
scientific article