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arXiv · 2602.19869

Optimal design with uncertainties: a risk-averse approach

Amal Alphonse·Petar Kunštek·Marko Vrdoljak
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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.

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Amal Alphonse, Petar Kunštek, Marko Vrdoljak. 2026-02-23. Optimal design with uncertainties: a risk-averse approach. https://arxiv.org/abs/2602.19869

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