**Abtract:**

Turbulence is an important example of chaotic dynamical systems, and sensitivity analysis is a powerful tool for design via computation. However, traditional sensitivity methods explode for chaotic systems, posing the challenge we will address in this talk. We first review how to compute meaningful derivatives of long-time-averaged objectives in chaotic systems via the shadowing method, which we then reformulate as a 'non-intrusive' minimization problem on the unstable subspace. Then we show a recent adjoint shadowing theorem, based on which we develop an adjoint sensitivity algorithm, NI... Read More