Preconditioned MINRES Method For Elliptic PDE-constrained Optimization Problems

In recent years, many efforts have been devoted to the elliptic partial differential equation constrained optimization distributed control problem. Of which the most com-mon one is to discretize the partial differential equation first and then solve the resulting system of linear equations. MINRES and GMRES methods have been applied to solve the resulting system of linear equations in literatures. In this paper, we proposed a parameter-ized block-diagonal preconditioned system and a block-counter-triangular preconditioned system which are solved respectively by preconditioned MINRES method and precondi-tioned GMRES method. The spectral analysis shows that the spectral distribution of the parameterized precondition matrix should be much more clustered if the parameter is greater than 1. We computed the eigenvalues and eigenvectors for block-counter-tridiagonal preconditioning matrix. Experiments show that the proposed preconditioned MINRES method and preconditioned GMRES method improve the computing speed ef-fectively for solving the elliptic PDE-constrained optimization problems.