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

Abstract

We analyse 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 adjoint 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

Dynamic Strategies for Yaw and Induction Control of Wind Farms Based on Large-Eddy Simulation and Optimization

“…For a more elaborate explanation and previous applications of the SP-Wind framework we refer the reader to Refs. [11,12,14,34,35].…”

Multigrid optimization for DNS-based optimal control in turbulent channel flows

“…In this context, the use of direct numerical simulation (DNS) or large-eddy simulation (LES) of turbulent flows implies large computational resources associated with the numerical evaluation of the objective functional that is used to define the optimization problem. Recently, various gradient based optimization algorithms have been tested for DNS-based optimal control in terms of number of forward and adjoint simulations required for cost functional improvement [14]. However, these methods were based on a single-grid optimization approach.…”

“…As a single-grid reference method, we employ a damped L-BFGS quasi-Newton method, which emerged in Ref. [14] as the most efficient single-grid method for the optimal-control problems that we consider in the current study.…”

“…We consider two different objective functionals in the optimal control problem, similar to the earlier work in Ref. [14]. The first objective functional aims at decreasing the turbulent kinetic energy at the end of the optimal control time horizon, which is relevant for achieving drag reduction [1,9].…”

Optimal Coordinated Control of Power Extraction in LES of a Wind Farm with Entrance Effects

“…However, given the current optimal control time span (approximately 2000 seconds), the noise levels of the statistical averages remain too high to find significant correlations. Ongoing work is focusing on improving the parallelization of the adjoint equations and the speed-up of our optimization algorithms [45,46], so that in the future, longer time averaging becomes possible. …”