Optimal design with uncertainties: a risk-averse approach

TL;DR

This paper develops a risk-averse optimal design framework for elliptic PDEs with uncertain coefficients, using homogenization, stochastic expansions, and a CVaR-based criterion, demonstrated through numerical examples.

Contribution

It introduces a novel risk-averse optimization approach incorporating uncertainty via CVaR, with theoretical existence results and an efficient numerical algorithm.

Findings

Existence of relaxed optimal designs established.

First-order optimality conditions derived.

Numerical method successfully applied to CVaR-based compliance minimization.

Abstract

We study a class of stochastic optimal design problems for elliptic partial differential equations in divergence form, where the coefficients represent mixtures of two conducting materials. The objective is to minimize a generalized risk measure of the system response, incorporating uncertainty in the loading through probability distributions. We establish existence of relaxed optimal designs via homogenization theory and derive first-order stationarity conditions satisfied by the optima. Based on these conditions, we develop an optimality criteria algorithm for numerical computations. The stochastic component is treated using a truncated Karhunen--Lo\`eve expansion, allowing evaluation of the value-at-risk (VaR) and conditional value-at-risk (CVaR) contributions arising from the sensitivity analysis and featured in the algorithm. The method is illustrated for an example involving CVaR-based compliance minimization.