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Optimal design with uncertainties: a risk-averse approach

Amal Alphonse, Petar Kunštek, Marko Vrdoljak

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--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.

Optimal design with uncertainties: a risk-averse approach

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--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.
Paper Structure (6 sections, 7 theorems, 85 equations, 2 figures, 1 algorithm)

This paper contains 6 sections, 7 theorems, 85 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. H-convergence and existence of solutions
  3. Necessary conditions of optimality
  4. Numerics via the optimality criteria method
  5. An example involving the conditional value-at-risk
  6. Conclusion

Key Result

Lemma 2.2

For ${\mathbf{A}} \in {\cal M}(\alpha,\beta;\Omega)$, the problem ses is uniquely solvable and $u_i \in L^p(S; H^1_0(\Omega))$ for $i = 1, \ldots, m$. Furthermore, we have the following a.s. estimate:

Figures (2)

  • Figure 1: Numerical solutions for $\eta=0.5$, $0.1$, and $0.05$ (displayed left to right, top to bottom) for ${\cal R} = {\mathrm{CVaR}_\gamma}$, along with the solution corresponding to the unperturbed right-hand side.
  • Figure 2: (Risk-neutral case) Numerical solutions for $\eta=0.5$, $0.05$ (displayed left to right) for ${\cal R} = \mathbb{E}$, along with the solution corresponding to the unperturbed right-hand side.

Theorems & Definitions (16)

  • Example 1.1
  • Remark 1.2
  • Definition 2.1: $H$-convergence
  • Lemma 2.2
  • proof
  • Theorem 2.5
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 6 more