An offline/online procedure for dual norm calculations of parameterized functionals: empirical quadrature and empirical test spaces

TL;DR

The paper introduces an efficient offline/online method for calculating the dual norm of parameterized functionals using empirical test spaces and quadrature, reducing computational costs in residual-based error estimation.

Contribution

It develops a novel approach combining empirical test spaces and quadrature for dual norm computation of parameterized functionals, addressing non-affine terms efficiently.

Findings

Effective reduction of offline computational costs.

Significant online efficiency improvements.

Validated on residual indicator computations.

Abstract

We present an offline/online computational procedure for computing the dual norm of parameterized linear functionals. The key elements of the approach are (i) an empirical test space for the manifold of Riesz elements associated with the parameterized functional, and (ii) an empirical quadrature procedure to efficiently deal with parametrically non-affine terms. We present a number of theoretical results to identify the different sources of error and to motivate the technique. Finally, we show the effectiveness of our approach to reduce both offline and online costs associated with the computation of the time-averaged residual indicator proposed in [Fick, Maday, Patera, Taddei, Journal of Computational Physics, 2018 (accepted)].