Reduced Basis Methods Based Upon Adaptive Snapshot Computations

2014/07/07 by Mazen Ali, Ali, Mazen, Kristina Steih +3
Engineering · Physics and Astronomy · #35B10 #41A30 #41A63 #65Y20 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1407.1708

openalex publication_date 2014/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB `truth space', but allow for error estimation of the RB approximation with respect to the exact solution of the considered parameterized partial differential equation. The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual. The resulting adaptive RBM is analyzed. We show the convergence of the resulting adaptive Greedy method. Numerical experiments for stationary and instationary problems underline the potential of this approach.

Citations

• Some observations on Babu\vska and Brezzi theories

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