On the efficiency of gradient based optimization algorithms for DNS-based optimal control in a turbulent channel flow

We analyze the performance of different limited-memory quasi-Newton methods for unconstrained DNS based optimization. Optimization based on Direct Numerical Simulation (DNS) of turbulent flows is extremely expensive, as functional and gradient evaluations require the simulation of Navier-Stokes and ad-joint Navier-Stokes equations with high space and time resolution. Nowadays, simple and robust nonlinear conjugate gradient methods are generally used for DNS-based optimal control, as they do not require much memory overhead in a large control space. In the current study, we investigate the use of quasi-Newton methods instead. They combine a cheap approximation of the Hessian to improve step direction and step length, leading to faster convergence of the optimization. Since control spaces are often large in DNS-based optimization, we investigate only limited-memory quasi-Newton methods. Three methods are studied, i.e., the discrete truncated Newton method, the limited-memory BFGS method, and the damped L-BFGS method. The latter method is designed for constrained optimization, but can also address unconstrained problems. Furthermore, the damped L-BFGS method only requires the Armijo condition in the line search, not the Wolfe conditions, limiting expensive functional and gradient evaluations. We investigate the combination of the three quasi-Newton methods with three different line-search methods either based on bisection, quadratic interpolation, or cubic interpolation. Initially, all possible combinations are evaluated in a test problem that is based on the extended Rosenbrock function. The three best performing methods are further tested in two different DNS-based optimal control cases in a turbulent channel flow at Re-tau = 180. This reveals that the damped L-BFGS method in combination with a cubic line search performs best, closely followed by classical L-BFGS with cubic line search. Though the damped L-BFGS often requires a few more iterations to reach convergence, this is compensated by a more cost effective line search, with fewer functional and gradient evaluations. Moreover, compared to the conjugate-gradient method, damped L-BFGS speeds up convergence by a factor of four. (C) 2015 Elsevier Ltd. All rights reserved.