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

Abstract

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, Maday, Patera, Taddei, Journal of Computational Physics, 2018].

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

“…The approach relies on a general (i.e., independent of the underlying PDE model) data compression procedure: given the snapshot set, we first perform space-time registration to "freeze" the position of the shock; then, we resort to POD to approximate the registered (mapped) field. To estimate the registered field, we resort to an hyper-reduced approximate minimum residual formulation: our statement is based on the introduction of a low-dimensional empirical test space [45] and of an empirical quadrature rule [16,54] to reduce online assembling costs.…”

“…, and G ∈ R , e , b ∈ R are a suitable matrix and vector, whose explicit expressions can be derived exploiting the same argument as in [16,45,54]. The problem of finding eq can thus be reformulated as a sparse-representation (or best-subset selection) problem:…”

“…Given a new value of the parameter ∈ , we resort to kernel-based regression to estimate the mapping coefficientŝ︀ a , while we resort to a space-time minimum residual projection-based ROM to compute the solution coefficientŝ︀ in (1.1). To reduce the costs of the minimum residual ROM, we first introduce a -dimensional empirical test space [45] and then we resort to an empirical quadrature procedure (EQP, [54]) to reduce the online assembling and memory costs. In Appendix C, we present mathematical justifications for linear problems of the procedure used to construct the test space ; we further present numerical results to illustrate the superiority of minimum residual ROMs compared to Galerkin ROMs.…”

A Registration Method for Model Order Reduction: Data Compression and Geometry Reduction

“…The EIM procedure can be extended to vector-valued fields. We present below the non-interpolatory extension of EIM employed in this paper; the same approach has also been employed in [53]. We refer to [55,36] for two alternatives applicable to vector-valued fields.…”

A projection-based model reduction method for nonlinear mechanics with internal variables: application to thermo-hydro-mechanical systems

“…The indicator ( 16) is expensive to evaluate since it relies on hf quadrature and it requires the computation of the supremum over all elements of X hf,0 : following [28], we consider the hyper-reduced error indicator…”