  This work investigates an elliptic optimal control problem defined on
uncertain domains and discretized by a fictitious domain finite element method
and cut elements. Key ingredients of the study are to manage cases considering
the usually computationally "forbidden" combination of poorly conditioned
equation system matrices due to challenging geometries, optimal control
searches with iterative methods, slow convergence to system solutions on
deterministic and non--deterministic level, and expensive remeshing due to
geometrical changes. We overcome all these difficulties, utilizing the
advantages of proper preconditioners adapted to unfitted mesh methods, improved
types of Monte Carlo methods, and mainly employing the advantages of embedded
FEMs, based on a fixed background mesh computed once even if geometrical
changes are taking place. The sensitivity of the control problem is introduced
in terms of random domains, employing a Quasi-Monte Carlo method and
emphasizing a deterministic target state. The variational discretization
concept is adopted, optimal error estimates for the state, adjoint state, and
control are derived that confirm the efficiency of the cut finite element
method in challenging geometries. The performance of a multigrid scheme
especially developed for unfitted finite element discretizations adapted to the
optimal control problem is also tested. Some fundamental preconditioners are
applied to the arising sparse linear systems coming from the discretization of
the state and adjoint state variational forms in the spatial domain. The
corresponding convergence rates along with the quality of the prescribed
preconditioners are verified by numerical examples.
