A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations

TL;DR

This paper introduces a numerically stable a posteriori error estimator for reduced basis methods applied to elliptic equations, improving accuracy and stability in model order reduction.

Contribution

The authors develop a new residual representation algorithm using an orthonormal basis, enhancing numerical stability and ease of implementation for RB error estimation.

Findings

The new estimator is more stable than standard residual-based methods.

It requires minimal additional computational effort.

Numerical tests demonstrate improved accuracy and stability.

Abstract

The Reduced Basis (RB) method is a well established method for the model order reduction of problems formulated as parametrized partial differential equations. One crucial requirement for the application of RB schemes is the availability of an a posteriori error estimator to reliably estimate the error introduced by the reduction process. However, straightforward implementations of standard residual based estimators show poor numerical stability, rendering them unusable if high accuracy is required. In this work we propose a new algorithm based on representing the residual with respect to a dedicated orthonormal basis, which is both easy to implement and requires little additional computational overhead. A numerical example is given to demonstrate the performance of the proposed algorithm.