Multigrid second-order accurate solution of parabolic control-constrained problems

Abstract

A mesh-independent and second-order accurate multigrid strategy to solve control-constrained parabolic optimal control problems is presented. The resulting algorithms appear to be robust with respect to change of values of the control parameters and have the ability to accommodate constraints on the control also in the limit case of bang-bang control. Central to the development of these multigrid schemes is the design of iterative smoothers which can be formulated as local semismooth Newton methods. The design of distributed controls is considered to drive nonlinear parabolic models to follow optimally a given trajectory or attain a final configuration. In both cases, results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed schemes are able to solve parabolic optimality systems with textbook multigrid efficiency. Further results are presented to validate second-order accuracy and the possibility to track a trajectory over long time intervals by means of a receding-horizon approach.

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

“…We consider second-order discretization schemes discussed in [10]. These schemes use second-order backward differentiation formula (BDF2) together with the Crank-Nicolson (CN) method in order to obtain a second-order time discretization scheme of the optimality system.…”

“…This is a nonlinear problem that includes an inequality constraint. To solve this problem, we adapt the scheme proposed in [10] to the case of state-constrained problems. This scheme is constructed by using a projection procedure to satisfy the inequality constraint (4.2c).…”

“…These problems require the development of algorithms that are fast and robust with respect to the optimization parameters. Recent developments [1][2][3]10] show that a successful framework to develop such algorithms is represented by space-time collective-smoothing multigrid schemes. In fact, Fourier analysis estimates [6,10] and results of numerical experiments with linear [1] and nonlinear [3,4,10] parabolic control problems demonstrate that space-time multigrid schemes provide optimal control solutions with meshindependent convergence and robustness with respect to the value of the control parameters.…”

“…Previous contributions to the multigrid solution of parabolic control problems [1,4,6,8,12] have focused on first-order time discretization, while higher-order space-time discretization and constraints on the control have been considered in [10]. In this contribution, the authors found that the Crank-Nicolson scheme is not a convenient choice while multistep backward differencing schemes are advantageous in the design of very efficient pointwise and linewise smoothers.…”

“…In this paper, we contribute to the field of space-time multigrid methods for the case of parabolic optimal control problems with state-constraints and second-order space-time discretization. Our multigrid approach is formulated based on criteria proposed in [1,2,10] combining space-time collective smoothing multigrid schemes and Lavrentiev regularization [21,25]. In this framework, the smoothing procedures are pointwise iterative schemes that update the optimization variables collectively and use projection to satisfy the inequality constraints.…”