Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems

Abstract

Solving parabolic PDE-constrained optimization problems requires to take into account the discrete time points all-at-once, which means that the computation procedure is often time-consuming. It is thus desirable to design robust and analyzable parallel-in-time (PinT) algorithms to handle this kind of coupled PDE systems with opposite evolution directions. To this end, for two representative model problems which are, respectively, the time-periodic PDEs and the initial-value PDEs, we propose a diagonalization-based approach that can reduce dramatically the computational time. The main idea lies in carefully handling the associated time discretization matrices that are denoted by Bper and Bini for the two problems. For the first problem, we diagonalize Bper directly and this results in a direct PinT algorithm (i.e., non-iterative). For the second problem, the main idea is to design a suitable approximation B̂per of Bini, which naturally results in a preconditioner of the discrete KKT system. This preconditioner can be used in a PinT pattern, and for both the Backward-Euler method and the trapezoidal rule the clustering of the eigenvalues and singular values of the preconditioned matrix is justified. Compared to existing preconditioners that are designed by approximating the Schur complement of the discrete KKT system, we show that the new preconditioner leads to much faster convergence for certain Krylov subspace solvers, e.g., the GMRES and BiCGStab methods. Numerical results are presented to illustrate the advantages of the proposed PinT algorithm.