8. Offline Error Bounds for the Reduced Basis Method

The reduced basis method is a model order reduction technique that is specifically designed for parameter-dependent systems. Due to an offline-online computational decomposition, the method is particularly suitable for the many-query or real-time contexts. Furthermore, it provides rigorous and efficiently evaluable a posteriori error bounds, which are used offline in the greedy algorithm to construct the reduced basis spaces and may be used online to certify the accuracy of the reduced basis approximation. Unfortunately, in real-time applications a posteriori error bounds are of limited use. First, if the reduced basis approximation is not accurate enough, it is generally impossible to go back to the offline stage and refine the reduced model; and second, the greedy algorithm guarantees a desired accuracy only over the finite parameter training set and not over all points in the admissible parameter domain. Here, we propose an extension or “add-on” to the standard greedy algorithm that allows us to evaluate bounds over the entire domain, given information for only a finite number of points. Our approach employs sensitivity information at a finite number of points to bound the error and may thus be used to guarantee a certain error tolerance over the entire parameter domain during the offline stage. We focus on an elliptic problem and provide numerical results for a thermal block model problem to validate our approach.