Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems

Abstract

In this paper, we describe block matrix algorithms for the iterative solution of large scale linear-quadratic optimal control problems arising from the optimal control of parabolic partial differential equations over a finite control horizon. We describe three iterative algorithms. The first algorithm employs a CG method for solving a symmetric positive definite reduced linear system involving only the unknown control variables. This system can be solved using the CG method, but requires double iteration. The second algorithm is designed to avoid double iteration by introducing an auxiliary variable. It yields a symmetric indefinite system and a positive definite block preconditioner. The third algorithm uses a symmetric positive definite block diagonal preconditioner for the saddle point system and is based on the parareal algorithm. Theoretical results show that the preconditioned algorithm has optimal convergence properties and parallel scalability. Numerical experiments are provided to confirm the theoretical results.

Convergence Analysis for Three Parareal Solvers

“…This is principally because the contraction factor K(z, J) is very complicated. For the Parareal-Euler algorithm, a rigorous analysis can be found in a recent paper by Mathew, Sarkis, and Schaerer [24], where the authors proved that…”

“…It permits the computation of the solution later in time, before having fully accurate approximations at earlier times, while the global accuracy of the iterative process after a few iterations is comparable to that given by a sequential numerical method used on a fine discretization in time. Nowadays, this algorithm, as well as some other relevant algorithms [5,7,9,10,11,22,29,21], have been used in many fields by many researchers, such as molecular-dynamics simulations [2], morphological transformation simulations [15], structural (fluid) dynamics simulations [5,10], optimal control [6,20,24,23], Hamiltonian simulations [8,14], simulations of turbulent plasmas [26,27], fast computation of wave equations [7] and Volterra integral equations [17], etc. For a survey of parallel-in-time algorithms, see [16].…”

Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues

“…Suppose the initial error {e 0 n } satisfies max n V e 0 n ∞ = O (1) and the machine precision is 2 −53 ≈ 1.1 × 10 −16 . For SPD problem (i.e., θ = 0), it has been observed in many places, e.g., [23,31], that the Parareal-Euler algorithm converges rapidly with a convergence factor around 0.3, and thus we need approximately 30 iterations to reach the machine precision. Therefore, the just mentioned '10 % increment' implies that: compared to the iteration number required for solving the SPD problem, at most 3 additional iterations are needed when we are now solving a non-SPD problem with θ ≤ 0.7.…”

“…For example, when we choose for G the Backward-Euler method and for F the Trapezoidal rule, which leads to the simplest implicit parareal algorithm with order 2 of the converged solution, the convergence rate can be arbitrarily slow and therefore is far away from the prediction by 'ρ ≈ 0.3'. The first work which treats the real situation is given in [23], there the authors analyzed the convergence properties of the Parareal-Euler algorithm consisting of using the Backward-Euler method as both the coarse and fine propagators, and they proved that the property 'ρ ≈ 0.3' is preserved by this algorithm. Recently, we prove that such a beautiful property also holds for some other familiar choices of the F -propagator [31], such as F =TR/BDF2 (i.e., the ODEs solver ode23tb in Matlab), F =2nd-SDIRK and F =3rd-SDIRK.…”

“…Dai et al proposed the projection-based-parareal algorithm for solving the Hamiltonian systems [5,14] and the wave equations [4], which consists of projecting the values obtained by the correction scheme to some specified constant manifold, e.g., the energy manifold, in each iteration. Maday et al [21,22] and Sarkis et al [6,23] embedded the parareal algorithm into the numerical discretization of the parabolic PDE-constrained control and optimization, where the parareal algorithm is assigned to treat the most heavy part in the whole computation, e.g., generating good proconditioner. Parareal as well as its other variants [7,16,24,32], e.g., the so-called SDC-Pipelined-Parareal proposed by Minion et al [7,24], are also successfully used for Volterra integral equations [18], turbulent plasma simulations [25][26][27], thermoviscoplastic problems [2], highly oscillatory PDEs [17] and singularly perturbed ODEs [19], etc.…”