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 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)].