An Adjoint Method using Fully Implicit Runge-Kutta Schemes for Optimization of Flow Problems

Abstract

The fully discrete adjoint equations and corresponding adjoint method are derived for unsteady PDE-constrained optimization problems. Specifically, we consider conservation laws on deforming domains that are temporally discretized by high-order fully implicit Runge-Kutta (IRK) schemes. Through a change of variables, the linear systems arising in the primal and dual problem are transformed, leading to computationally cheaper systems that compare competitively with those derived from diagonally implicit Runge-Kutta (DIRK) schemes. Quantities of interest that take the form of space-time integrals are discretized in a solver-consistent manner. Our fully discrete, IRK adjoint method is used to compute exact gradients of quantities of interest with respect to the optimization parameters. These quantities of interest and their gradients are used for gradient-based PDE-constrained optimization. Our implementation of this IRK adjoint method is tested by computing the energetically optimal trajectory of a 2D airfoil in flow governed by the compressible Navier-Stokes equations. We also analyze the parallel performance of our IRK adjoint method and the DIRK adjoint method, showing that our implementation is computationally comparable.

A robust, discrete-gradient descent procedure for optimisation with time-dependent PDE and norm constraints

“…When impractical to use AD for some of the reasons mentioned above, the alternative approach is to derive analytically the discrete adjoint equations using summation by parts. This approach, albeit established in the optimisation literature [20,61], has been less discussed until recently in the aeronautics [18,60] and fluid dynamics literature [14,30,51,57].…”

Discrete adjoint-based control: A robust gradient descent procedure for optimisation with PDE and norm constraints

“…While this approach works well on codes written so as to be amenable to AD such as dolfin-adjoint [19,20], it can otherwise consume large amounts of memory and be extremely challenging to debug [21,22]. The alternative approach is to derive and code the discrete adjoint equations, an established approach in the optimisation literature [23,24], but until recently less discussed in the aeronautics [25,26] and fluid dynamics literature [27,28,29,30].…”