Optimization Algorithm for Systems Governed by Chaotic Dynamics

We describe an algorithm for optimizing time-averaged objective functions that depend on a chaotic state variable. Such problems are ubiquitous in engineering design. They are challenging, because of the sensitive dependence of the state to perturbations in the design. One consequence of this sensitive dependence is that increasing the averaging period, which improves the accuracy of the objective, causes the gradient to diverge. To overcome this issue, the proposed algorithm uses an ensemble objective in a Newton-Krylov trust-region framework. The ensemble objective averages a set of objective functions, each of which uses a reduced time-averaging period and independent state variable; this independence permits each simulation and adjoint computation to be carried out in parallel. The novel aspect of the proposed method is the use of the ensemble objective within a Newton-Krylov algorithm; the latter helps avoid some of the issues presented by objectives governed by chaotic state variables. We demonstrate the proposed ensemble-Newton-Krylov algorithm on an optimization problem governed by the Lorenz dynamical system.