Abstract

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. The key concept of the RB method is to prepare a problem-adapted 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.

To assess the approximation error caused by the RB method in the online stage a reliable and efficient a posteriori error estimator has been derived in Veroy et al. (2003). However, for inf-sup stable problems such as acoustics problems the estimation of the inf-sup constant still poses a challenge and the existing methods often result in rather pessimistic results and thus pessimistic error bounds. We propose a constant-free, probabilistic a posteriori error estimator that does not require to estimate any stability constants and is both reliable and efficient at (given) high probability. Here, we extend the approach in Cao and Petzold (2004); Homescu et al. (2005), where the solution of an adjoint problem with random conditions at the final time is employed to estimate the approximation error for ordinary differential equations.