Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method

TL;DR

This paper improves the accuracy and efficiency of evaluating the a posteriori error estimator in the reduced basis method by employing the Empirical Interpolation Method (EIM) and addressing round-off errors, demonstrated on diffusion and acoustic scattering problems.

Contribution

It introduces an enhanced approximation of the error bound using EIM, reducing precomputations and stabilizing against round-off errors in reduced basis methods.

Findings

EIM-based approximation improves error bound accuracy.

The method reduces precomputational costs.

Stabilization against round-off errors is effective.

Abstract

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.