Adaptive Parameter Optimization For An Elliptic-Parabolic System Using The Reduced-Basis Method With Hierarchical A-Posteriori Error Analysis

TL;DR

This paper develops an adaptive reduced-basis method with hierarchical a-posteriori error analysis to efficiently optimize parameters in a nonlinear elliptic-parabolic system modeling lithium-ion batteries, demonstrating high accuracy with few basis functions.

Contribution

It introduces a novel adaptive reduced-basis approach with hierarchical error estimators for parameter optimization in nonlinear coupled systems, specifically applied to battery modeling.

Findings

Efficient parameter optimization with few basis functions.

Hierarchical a-posteriori error estimators improve reliability.

Numerical results show high accuracy and computational efficiency.

Abstract

In this paper the authors study a non-linear elliptic-parabolic system, which is 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. However, the states are coupled through a strongly non-linear 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. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced basis functions.