Reduced basis methods with adaptive snapshot computations

• Published in: Advances in computational mathematics Vol. 43; no. 2; pp. 257 - 294
• Main Authors: Ali, Mazen, Steih, Kristina, Urban, Karsten
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2017; Springer Nature B.V
• Subjects: Asymptotic methods; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Exact solutions; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical methods; Partial differential equations; Visualization; Adaptivity; Wavelets; 41A30; 41A63; 35B10; 65Y20; Reduced basis method; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-016-9485-9

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.