Saddle Point Problem In PDE Control Optimization Problem

Optimization problems with PDE constraints arise widely in many areas, such as sciences and engineering,computation dynamics fluids and electromag-netism. Here we consider2-dimensional Possion problems is the PDE, the large linear system results from it’s discretization is of saddle-point type. Tyrone Rees[,,] has done many works on solving the problem by preconditioned Krylov subspace method. We also use a symmetric indefinite preconditioner to solve this system. On the other hand,we can easily reduce the3×3block sys-tem to a2x2one,which is of stabilized type. However we realized that the block diagonally predictioned MINRES doesn’t work efficiently here when the regularization parameter β is small. We introduce two ideas to deal with this case, one is giving a new approximation of Schur complement and the other is turn the reduced system to a equivalence structure which has better properties. Numerical example demonstrates our ideas are pratical and efficient.