# A Goal-Oriented Adaptive Discrete Empirical Interpolation Method

## TL;DR

This paper introduces an adaptive DEIM algorithm and a-posteriori error estimation techniques to improve the accuracy of quantities of interest in reduced order models, validated on fluid dynamics models.

## Contribution

It develops a novel adaptive DEIM method combined with a-posteriori error estimation for enhanced reduced order model accuracy.

## Key findings

- Error estimates closely match theoretical predictions
- Adaptive DEIM improves quantity of interest accuracy
- Validated on 1D-Burgers and Shallow Water models

## Abstract

In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method. First order expansions of the components of the quantity of interest obtained as the product between the components gradient and model residuals are summed up to generate the error estimation result. Efficient versions are derived for explicit and implicit Euler schemes and require only one reduced forward and adjoint models and high-fidelity model residuals estimation. Then we derive an adaptive DEIM algorithm to enhance the accuracy of these quantities of interests. The adaptive DEIM algorithm uses dual weighted residuals singular vectors in combination with the non-linear term basis. Both the a-posteriori error estimation results and the adaptive DEIM algorithm were assessed using the 1D-Burgers and Shallow Water Equation models and the numerical experiments shows very good agreement with the theoretical results.