Stochastic Dominance Constraints in Elastic Shape Optimization

Abstract

This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The underlying cost functional is of compliance type plus a perimeter term, in the implementation shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations.

A pessimistic bilevel stochastic problem for elastic shape optimization

“…In the stochastic setting, we restrict ourselves to the expected value E [F[u]] as the risk measure for the optimization (cf. (10)). Furthermore, the stochastic perturbation of the distribution of the thickness parameter u is given by i.i.d.…”

“…The approach presented here is based on our previous work in [8][9][10], which grew out of the aspiration to mobilize methodology from mainly economy-driven decision making under (stochastic) uncertainty in order to study PDE-constrained optimization with an emphasis on engineering-related topics such as shape optimization. The riskneutral models and models with risk aversion in the objective or the constraints were treated with the classical expectation, with risk measures, or by invoking comparisons using stochastic dominance relations.…”

Fast and Efficient Numerical Finite Difference Method for Multiphase Image Segmentation

“…Oudet and Santambrogio [6] presented the Modica-Mortola approximation for branched transport. Conti et al [7] investigated elastic shape optimization. Pegon et al [8] studied fractal shape optimization in branched transport.…”

Stochastic approximation for optimization in shape spaces

“…Increasingly, stochastic models are being used in shape optimization with the goal of obtaining more robust solutions. A number of works has focused on structural optimization with either random Lamé parameters or forcing [3,10,11,13,36]. Stochastic models have also handled uncertainty in the geometry of the domain [7,25,33].…”

“…Stochastic models have also handled uncertainty in the geometry of the domain [7,25,33]. To ensure well-posedness of the stochastic problem, either an order must be defined on the relevant random variables, as in [11], or the problem needs to be transformed to a deterministic one by means of a probability measure. One possibility is to compute the worst case design [4,14].…”