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Towards Parallel-in-time Multiple Shooting Methods for Large-scale Optimal Control Problems governed by the Navier–Stokes Equations

In recent decades, the study of optimal control problems governed by Navier–Stokes equations has received noticeable attention in many applications. In the turbulent regime, the model equations in the control problem correspond to direct numerical simulations or large-eddy simulations, making the solution of the underlying optimization problem very expensive and challenging. Recent work has focused on improving optimization algorithms for these types of problems. Therefore, the main objective of this work is to propose a more efficient and parallel-in-time optimization algorithm for turbulence control problems.

In the current work, we apply the multiple shooting method to the large-scale PDE governed systems. Multiple shooting methods for solving optimal control problems have been developed rapidly in the past decades and are widely considered a promising direction to speed up the optimization process. However, their application for solving large-scale PDE-based optimal control problems still faces many challenges, including the difficulty of solving large-scale equality constrained optimization problems in an efficient parallelizable way.

To address these issues, we solve the equality constrained optimization problems introduced by the multiple shooting strategy by using the augmented Lagrangian (AL) method, in which the unconstrained subproblems are solved using a classical limited-memory BFGS quasi-Newton method. An optimal control problem governed by the Nagumo equation is employed to validate the proposed algorithm and analyze its efficiency. The results demonstrate that substantial accelerations can be achieved for multiple shooting approaches when proper starting guesses of controls are provided, and when the control variables are scaled appropriately. A second test case consists of a two-dimensional velocity tracking problem that is governed by the Navier–Stokes equations. The influence of the flow complexity on the optimization method is studied, and the results illustrate that for a fluid field with more complex structures, the efficiency of the algorithm further increases. Overall, for the different cases considered, we find algorithmic speed-ups of the proposed AL-based multiple shooting algorithm of up to 6 versus single shooting, depending on the starting guess, and the specific tracking problem.

Furthermore, we propose a new multiple shooting algorithm based on a sequential quadratic programming (SQP) method. The structure of the KKT matrix is investigated and the corresponding large-scale KKT system is solved by a preconditioned conjugate gradient algorithm. The Hessian and its inverse are approximated by the limited-memory BFGS method. A simplified block Schur complement preconditioner is proposed, that allows for the parallelization of the method in the time domain. Finally, a line search algorithm using the L1-merit function with the watchdog strategy is employed to ensure global convergence. The proposed SQP algorithm is first validated on the Nagumo problem. The results also indicate that considerable accelerations can be achieved for multiple shooting approaches with appropriate starting guesses and scaling of the matching conditions. We further apply the proposed algorithm to the 2D velocity tracking problem governed by the Navier–Stokes equations. We find algorithmic speed-ups of up to 12 versus single shooting on up to 50 shooting windows. We also compare results with earlier work that uses an augmented Lagrangian algorithm instead of SQP, showing better performance of the SQP method for most of the cases.

Date:2 Sep 2016 →  24 Mar 2022
Keywords:Turbulent flow, Chaotic optimization
Disciplines:Electrical power engineering, Energy generation, conversion and storage engineering, Thermodynamics, Mechanics, Mechatronics and robotics, Manufacturing engineering, Safety engineering
Project type:PhD project