A hierarchical a posteriori error estimator for the Reduced Basis Method

• Published in: Advances in computational mathematics Vol. 45; no. 5-6; pp. 2191 - 2214
• Main Authors: Hain, Stefan, Ohlberger, Mario, Radic, Mladjan, Urban, Karsten
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2019; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Decay rate; Error analysis; Galerkin method; Greedy algorithms; Lower bounds; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Model reduction; Model reduction of parametrized Systems; Nonlinear programming; Parameters; Tightness; Visualization; Hierarchical error estimator; 65M15; 65N15; A posteriori error estimator; Reduced Basis Method; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09675-z

In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of inf - sup stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N -with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the inf - sup constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent inf - sup constant—may become infeasible.