PinT PRECONDITIONER FOR FORWARD-BACKWARD EVOLUTIONARY EQUATIONS

Solving the linear system (kappa kappa(T))(-1) b is often the major computational burden when a forward-backward evolutionary equation must be solved in a problem, where kappa is the so-called all-at-once matrix of the forward subproblem after space-time discretization. An efficient solver requires a good preconditioner for kappa kappa(T). Inspired by the structure of kappa, we precondition kappa kappa(T) by P alpha P alpha T being P-alpha block alpha-circulant matrix constructed by replacing the Toeplitz matrices in kappa by the alpha-circulant matrices. By a block Fourier diagonalization of P-alpha, the computation of the preconditioning step (P alpha P alpha T )(-1) r is parallelizable for all the time steps. We give a spectral analysis for the preconditioned matrix (P alpha P alpha T )(-1) (kappa kappa(T)) and prove that for any one-step stable time-integrator the eigenvalues of (P alpha P alpha T )(-1) (kappa kappa(T)) spread in a mesh-independent interval [(1+root 2 delta)(-1), (1-root 2 delta)(-1)] if the parameter alpha weakly scales in terms of the number of time steps N-t as alpha = delta/root N-t, where delta is an element of (0, 1/root 2) is a free constant. Two applications of the proposed preconditioner are illustrated: PDE-constrained optimal control problems and parabolic source identification problems. Numerical results for both problems indicate that spectral analysis predicts the convergence rate of the preconditioned conjugate gradient method very well.