Taylor approximation for PDE‐constrained optimization under uncertainty: Application to turbulent jet flow

We present a scalable method for PDE‐constrained optimization under high‐dimensional uncertainty. The method is based on a functional Taylor approximation of the cost functional with respect to (w.r.t.) the random parameter field. We consider a risk‐averse optimization formulation via a mean‐variance measure for the control objective. A quadratic Taylor approximation of the control objective gives rise to a function involving the trace of the Hessian of the objective w.r.t. the random parameter, preconditioned by the covariance operator. We apply a randomized SVD algorithm to estimate the trace, which is demonstrated to be scalable w.r.t. the parameter dimension, i.e., the number of PDE solves is independent of the parameter dimension and depends only on the decay of the eigenvalues of the covariance‐preconditioned Hessian (which is typically dimension‐independent). We solve the resulting generalized eigenproblem‐constrained optimization problem by a quasi‐Newton method. We apply our method to a RANS turbulence model with an infinite‐dimensional random field representing model inadequacy, and demonstrate scalability on discretized problems with parameter dimensions up to one million.