Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

Abstract

Optimization with time-dependent partial differential equations (PDEs) as constraints appears in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward time-dependent PDE coupled with optimality conditions. An optimization process by using the one-shot method determines the optimal control, state and adjoint state at once, with the cost of solving a large scale, fully discrete optimality system. Hence, such a one-shot method could easily become computationally prohibitive when the time span is long or time step is small. To overcome this difficulty, we propose several time domain decomposition algorithms for improving the computational efficiency of the one-shot method. In these algorithms, the optimality system is split into many small subsystems over a much smaller time interval, which are coupled by appropriate continuity matching conditions. Both one-level and two-level multiplicative and additive Schwarz algorithms are developed for iteratively solving the decomposed subsystems in parallel. In particular, the convergence of the one-level, non-overlapping algorithms is proved.The effectiveness of our proposed algorithms is demonstrated by both 1D and 2D numerical experiments, where the developed two-level algorithms show convergence rates that are scalable with respect to the number of subdomains.

A Scalable Space-Time Domain Decomposition Approach for Solving Large Scale Nonlinear Regularized Inverse Ill Posed Problems in 4D Variational Data Assimilation

“…A nonintrusive framework for integrating existing unsteady PDE solvers into a parallel-in-time simultaneous optimization algorithm, using PFASST, is provided in [23]. Related parallel PDE solvers based on a Schwarz preconditioner in space-time are proposed in [20,30,54].…”

A splitting algorithm for constrained optimization problems with parabolic equations

“…Therefore, there exist fruitful research results on this topic theoretically and numerically [4,34], especially for the elliptic optimal control problem [15,16,32]. Because of the large scale of discrete system and the limitation of computational resources, the design of the high accuracy mathematical scheme for parabolic optimal control problem is difficult, and we refer the readers to [1,18,19,20,21,22,23] and references therein. Based on the finite element approximation, the Crank-Nicolson scheme, the numerical integration formula, and the full Jacobian decomposition method with correction [1,17], an efficient numerical algorithm is proposed for the parabolic optimal control problem in this paper.…”

“…Based on the traditional preconditioning techniques for saddle-point systems that arise from static PDE constrained optimal control problems ( [26,27,29,33]), some preconditioning methods are proposed for the time evolution problems, we refer to [5,10,19,30] and references therein for the rich literature. Moreover, there also exist some parallel algorithms for solving the optimal control problem (1.1)-(1.2) based on above two approaches, such as the time domain decomposition methods [20,28], and the non-intrusive parallel-in-time approach [14]. The efficiency of these parallel methods have been proved, but the implements are complicated.…”