Multigoal-oriented optimal control problems with nonlinear PDE constraints
This work provides a method for balancing discretization and nonlinear iteration errors in multigoal-oriented optimal control, which is incremental for researchers in PDE-constrained optimization.
The authors developed an a posteriori error representation and mesh adaptivity for multigoal-oriented optimal control problems with nonlinear PDE constraints, applied to the regularized p-Laplace equation. Numerical examples demonstrate the effectiveness of the adaptive solution strategy.
In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples.