Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters

Abstract

We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical computational challenges of expensive PDE solution and high-dimensional uncertainty, we construct surrogates of the constraint function by Taylor approximation, which relies on efficient computation of the derivatives, low rank approximation of the Hessian, and a randomized algorithm for eigenvalue decomposition. To tackle the difficulty of the non-differentiability of the inequality chance constraint, we use a smooth approximation of the discontinuous indicator function involved in the chance constraint, and apply a penalty method to transform the inequality constrained optimization problem to an unconstrained one. Moreover, we design a gradientbased optimization scheme that gradually increases smoothing and penalty parameters to achieve convergence, for which we present an efficient computation of the gradient of the approximate cost functional by the Taylor approximation. Based on numerical experiments for a problem in optimal groundwater management, we demonstrate the accuracy of the Taylor approximation, its ability to greatly accelerate constraint evaluations, the convergence of the continuation optimization scheme, and the scalability of the proposed method in terms of the number of PDE solves with increasing random parameter dimension from one thousand to hundreds of thousands.

Optimality Conditions and Moreau--Yosida Regularization for Almost Sure State Constraints

“…The framework presented in our work is general enough to cover the theory in [17] without additional structure on the probability space. It is worth mentioning that our work related to the recent trend to include chance constraints in PDE-constrained optimization under uncertainty [12,16,20]. These models are of interest when deviations from a hard constraint are permissible with a certain probability.…”