Optimal design with uncertainties: a risk-averse approach
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910030203191296 |
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| author | Alphonse, Amal Kunštek, Petar Vrdoljak, Marko |
| author_facet | Alphonse, Amal Kunštek, Petar Vrdoljak, Marko |
| contents | 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--Loève 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal design with uncertainties: a risk-averse approach Alphonse, Amal Kunštek, Petar Vrdoljak, Marko Optimization and Control Analysis of PDEs 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--Loève 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. |
| title | Optimal design with uncertainties: a risk-averse approach |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2602.19869 |