A hierarchical a posteriori error estimator for the Reduced Basis Method

Abstract

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 weak 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.

Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations

“…Additionally, they can be challenging to implement, as they require Lipschitz or inf-sup constants to be estimated or bounded, which incurs additional computational cost and can further degrade the bounds' sharpness [23,24]. In the reduced-basis context, a recently proposed hierarchical error estimator can yield sharper estimates without the need to compute these constants, at the cost of solving a higher-dimensional reduced-order model [25]. Finally, these (deterministic) bounds are of limited utility in an uncertainty-quantification setting, where a statistical model of the error is more readily integrable into the uncertainty analysis.…”

A New Error Estimator for Reduced-Order Modeling of Linear Parametric Systems

“…which are used to construct the reduced systems in (5), (7), (14) or in (23), (33), respectively. For simplicity and clarity of analysis, we only use Galerkin projection for all the reduced systems, so that only one projection matrix V, V du , V r du or V rpr , V rrpr needs to be computed for each reduced system.…”

“…Another error estimation which is independent of the inf-sup constant is proposed in [7]. This error estimation is used to estimate the error of the state (solution vector).…”

“…As for estimation of the transfer function error or output error, trivially multiplying the output matrix norm C with the error estimator could also estimate the output error, but may lead to slow error decay if C is large. Moreover, a saturation constant needs to be estimated for the error estimator in [7], which needs extra computations and may cause inefficiency of the error estimator if computed without sufficient accuracy.…”

Randomized Residual-Based Error Estimators for Parametrized Equations

“…For instance, the Successive Constraint Method (SCM) [17,5,16] computes a parameter-dependent lower bound of the infsup constant by employing the successive solution to appropriate linear optimization problems. This procedure is usually computationally demanding and can lead to pessimistic error bounds [12].…”