The paper addresses the challenge of efficiently solving norm-constrained, nonlinear optimal control problems in fluid dynamics. These problems are crucial in various applications, such as improving transport and mixing efficiencies in complex fluids, analyzing the stability of thermo-acoustic oscillations, modeling transitions between multiple basins of attraction in dynamical systems, and forecasting oceanographic systems. Due to the nonlinear nature of the governing partial differential equations (PDEs) and the high computational cost of direct numerical simulations, efficient optimization routines are essential for tackling these problems.
The authors propose a numerically consistent, discrete formulation of the direct-adjoint looping method, accompanied by gradient descent and line-search algorithms with global convergence guarantees. They derive the discrete adjoint equations for time-dependent PDE constraints using multistep schemes and compare the performance of the discrete and continuous adjoint approaches on several example problems. Furthermore, the authors utilize tools from the optimization on manifolds community to develop a consistent and efficient gradient update procedure that restricts the control variable to a spherical manifold, eliminating the need for equality constraints.
The main findings of the manuscript demonstrate the superiority of the discrete adjoint approach in providing accurate gradient estimates and the effectiveness of the proposed gradient descent procedure in achieving robust convergence to stationary points with small residual errors. The authors test their methodology on three fluid dynamics problems: a dynamo problem, the Swift-Hohenberg multi-stability problem, and an optimal mixing problem. In all cases, the discrete adjoint formulation outperforms the continuous adjoint approach, and the gradient descent procedure exhibits linear to super-linear convergence rates.
The significance of this research lies in its contribution to the efficient solution of norm-constrained optimization problems in fluid dynamics. The proposed discrete adjoint formulation and gradient descent procedure offer a robust and computationally efficient framework for tackling a wide range of applications in the field. Moreover, the authors provide an accompanying library, SphereManOpt, which implements their methodology and can be readily used by the scientific community. This work has the potential to advance the understanding and optimization of complex fluid systems, with implications for various domains, such as oceanography, mixing processes, and stability analysis.