On Approximate Block Preconditioning for Discretized Optimal Control Problems Constrained with Space‐Fractional Diffusion Equations

For a class of optimal control problems constrained with certain space‐fractional diffusion equations, by making use of the right rectangular rule for the cost function and the implicit finite difference scheme with the shifted Grünwald formula for the constraint equation along with Lagrange multiplier approach, we obtain specially structured block two‐by‐two linear systems. We construct the circulant‐based and ‐matrix‐based approximate block preconditioning matrices for the coefficient matrices of the discrete linear systems and analyze spectral properties of the corresponding preconditioned matrices. Theoretical results indicate that except for a small number of outliers the eigenvalues of the preconditioned matrices are clustered around 1. Numerical experiments show that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace methods.