author = {Conti, Sergio and Rumpf, Martin and Schultz, R{\"u}diger and T{\"o}lkes,
  title = {Stochastic Dominance Constraints in Elastic Shape Optimization},
  journal = {SIAM Journal on Control and Optimization},
  note = {accepted},
  year = {2018},
  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.},
  arxiv = {},
  eprint = {1606.09461},
  pdf = { 1}