PinT Preconditioner for Forward-Backward Evolutionary Equations

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1771-1798, December 2023.
Abstract. Solving the linear system [math] is often the major computational burden when a forward-backward evolutionary equation must be solved in a problem, where [math] is the so-called all-at-once matrix of the forward subproblem after space-time discretization. An efficient solver requires a good preconditioner for [math]. Inspired by the structure of [math], we precondition [math] by [math] with [math] being a block [math]-circulant matrix constructed by replacing the Toeplitz matrices in [math] by the [math]-circulant matrices. By a block Fourier diagonalization of [math], the computation of the preconditioning step [math] is parallelizable for all the time steps. We give a spectral analysis for the preconditioned matrix [math] and prove that for any one-step stable time-integrator the eigenvalues of [math] spread in a mesh-independent interval [math] if the parameter [math] weakly scales in terms of the number of time steps [math] as [math], where [math] 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.