1 research outputs found
Random geometries and Quasi Monte Carlo methods for optimal control PDE problems based on fictitious domain FEMS and cut elements
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. 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.
Further, a practical error estimate for the Quasi--Monte Carlo method is
derived, showing that it yields the theoretically predicted convergence rate