PinT Preconditioner for Forward-Backward Evolutionary Equations

Solving the linear system \(({\mathcal{K}}{\mathcal{K}}^\top )^{-1}{\boldsymbol b}\) is often the major computational burden when a forward-backward evolutionary equation must be solved in a problem, where \(\mathcal{K}\) is the so-called all-at-once matrix of the forward subproblem after space-time discretization. An efficient solver requires a good preconditioner for \({\mathcal{K}}{\mathcal{K}}^\top\). Inspired by the structure of \(\mathcal{K}\), we precondition \({\mathcal{K}}{\mathcal{K}}^\top\) by \({\mathcal{P}}_\alpha{\mathcal{P}}_\alpha^\top\) with \({\mathcal{P}}_\alpha\) being a block \(\alpha\)-circulant matrix constructed by replacing the Toeplitz matrices in \(\mathcal{K}\) by the \(\alpha\)-circulant matrices. By a block Fourier diagonalization of \({\mathcal{P}}_\alpha\), the computation of the preconditioning step \(({\mathcal{P}}_\alpha{\mathcal{P}}_\alpha^\top )^{-1}{\boldsymbol r}\) is parallelizable for all the time steps. We give a spectral analysis for the preconditioned matrix \(({\mathcal{P}}_\alpha{\mathcal{P}}_\alpha^\top )^{-1}({\mathcal{K}}{\mathcal{K}}^\top )\) and prove that for any one-step stable time-integrator the eigenvalues of \(({\mathcal{P}}_\alpha{\mathcal{P}}_\alpha^\top )^{-1}({\mathcal{K}}{\mathcal{K}}^\top )\) spread in a mesh-independent interval \([(1+\sqrt{2}\delta )^{-1}, (1-\sqrt{2}\delta )^{-1}]\) if the parameter \(\alpha\) weakly scales in terms of the number of time steps \(N_t\) as \(\alpha =\delta/\sqrt{N_t}\), where \(\delta \in (0, 1/\sqrt{2})\) 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.