Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
For practitioners of reduced basis methods, this work improves the reliability and efficiency of error certification, though it is an incremental improvement over existing techniques.
The paper addresses round-off error sensitivity in the a posteriori error bound evaluation for the reduced basis method. It proposes an improved approximation using the Empirical Interpolation Method (EIM) that achieves higher accuracy and requires fewer precomputations, demonstrated on 1D diffusion and 3D acoustic scattering problems.
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.