Parameter optimization for elliptic-parabolic systems by an adaptive trust-region reduced basis method

Abstract

In this chapter the authors study parameter optimization problems for nonlinear elliptic-parabolic systems, which are motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced order approach based on the reduced basis (RB) method, where no (computationally expensive) offline phase is required. However, the states are coupled through a strongly nonlinear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization are described for the coupled system. For the parameter optimization an adaptive trust-region method is utilized, where the RB approximation is controlled by efficiently computable approximated hierarchical a-posteriori error estimators. Numerical experiments illustrate sufficiently good approximation properties and efficiencies by using only a relatively small number of RB functions.