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During the last decades (numerical) simulations based on partial differential equations (PDEs) have considerably gained importance in engineering applications, life sciences, environmental issues, and finance. However, especially when multiple simulation requests or a real-time simulation response are desired, standard methods such as finite elements (FE) are prohibitive. Model reduction approaches such as the reduced basis (RB) method, which we will consider here, have been developed to tackle such situations (see for instance Haasdonk (2017); Quarteroni et al. (2016); Hesthaven et al. (2016) for an overview). The key concept of the RB method is to prepare a problemadapted low-dimensional subspace of the high-dimensional (FE) discretization space in a possibly expensive offline stage to realize a fast simulation response by Galerkin projection on that low-dimensional space in the subsequent online stage.

Constant-Free, Randomized a Posteriori Error Estimators for Parameter-Dependent Partial Differential Equations