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We propose a fast and scalable optimization method to solve chance orprobabilistic constrained optimization problems governed by partialdifferential equations (PDEs) with high-dimensional random parameters. Toaddress the critical computational challenges of expensive PDE solution andhigh-dimensional uncertainty, we construct surrogates of the constraintfunction by Taylor approximation, which relies on efficient computation of thederivatives, low rank approximation of the Hessian, and a randomized algorithmfor eigenvalue decomposition. To tackle the difficulty of thenon-differentiability of the inequality chance constraint, we use a smoothapproximation of the discontinuous indicator function involved in the chanceconstraint, and apply a penalty method to transform the inequality constrainedoptimization problem to an unconstrained one. Moreover, we design agradient-based optimization scheme that gradually increases smoothing andpenalty parameters to achieve convergence, for which we present an efficientcomputation of the gradient of the approximate cost functional by the Taylorapproximation. Based on numerical experiments for a problem in optimalgroundwater management, we demonstrate the accuracy of the Taylorapproximation, its ability to greatly accelerate constraint evaluations, theconvergence of the continuation optimization scheme, and the scalability of theproposed method in terms of the number of PDE solves with increasing randomparameter dimension from one thousand to hundreds of thousands.

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