The numerical solution of large-scale risk-averse PDE-constrained optimization problems requires substantial computational effort due to the discretization in physical and stochastic dimensions. Managing the cost is essential to tackle such problems with high dimensional uncertainties. In this work, we combine an inexact trust-region (TR) algorithm from with a local, reduced basis (RB) approximation to efficiently solve risk-averse optimization problems with DE constraints. The main contribution of this work is a numerical framework for systematically constructing surrogate models for the TR subproblem and objective function using local sample-based approximations. Under standard assumptions, the inexact TR algorithm is guaranteed to converge from any initial guess, provided that errors in the evaluation of the objective function and its gradient using our RB approach are adequately bounded. In this work, we provide conditions for which our RB approximations satisfy the necessary bounds and demonstrate the performance of our proposed approach through numerical examples. These examples demonstrate that we can efficiently solve risk-averse PDE-constrained optimization problems with significant computational savings when compared to Monte Carlo.

Collaborators: Drew Kouri, Sandia National Laboratories and Zilong Zou, Duke University