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

We present an offline/online computational procedure for computing the dual norm of parameterized linear functionals. The approach is motivated by the need to efficiently compute residual dual norms, which are used in model reduction to estimate the error of a given reduced solution. 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 and numerical results to identify the different sources of error and to motivate the proposed technique, and we compare the approach with other state-of-the-art techniques. Finally, we investigate 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 et al. (J. Comput. Phys. 371, 214–243 2018).