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

Understand

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 gradient-based 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.