Dual natural-norm a posteriori error estimators for reduced basis approximations to parametrized linear equations

In this work, the concept of dual natural-norm for parametrized linear equations is used to derive residual-based a posteriori error bounds characterized by a O?(1) stability constant. We translate these error bounds into very effective practical a posteriori error estimators for reduced basis approximations and show how they can be efficiently computed following an offline/online strategy. We prove that our practical dual natural-norm error estimator outperforms the classical inf-sup based error estimators in the self-adjoint case. Our findings are illustrated on anisotropic Helmholtz equations showing resonant behavior. Numerical results suggest that the proposed error estimator is able to successfully catch the correct order of magnitude of the reduced basis approximation error, thus outperforming the classical inf-sup based error estimator even for non-self-adjoint problems.

Automated summary

In this work, dual natural-norm is utilized to derive a posteriori error bounds with a stability constant of O(1) for parametrized linear equations. These error bounds are then translated into effective practical error estimators for reduced basis approximations using an offline/online strategy. The practical dual natural-norm error estimator is proven to outperform the classical inf-sup based error estimators in the self-adjoint case. Numerical results demonstrate its ability to accurately estimate the reduced basis approximation error, even for non-self-adjoint problems, surpassing the classical inf-sup based error estimator.