Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

Abstract

The a posteriori error analysis of a reduced basis (RB) method may certify the accuracy of an RB solution without the truth finite element (FE) solution. However, the error theory necessitates a numerically expensive stability factor such as coercivity or an inf–sup factor. Consequently, we perforce resort to the successive constraint method (SCM) to achieve a stability lower bound for a rigorous albeit rapid RB analysis. As an alternative, we propose an adaptive Gaussian process (AGP) to produce a stability lower bound for the rigorous, rapid, and sharp estimation of an RB solution error. We utilize linear elastostatic and Helmholtz acoustic problems with geometric parameterization to compare the AGP with the SCM for a numerical investigation. The results show that the AGP significantly surpasses the SCM. This is because the a posteriori error estimation of an RB solution obtained with the AGP is sharper and faster than that obtained with the SCM by factors of at least 103 and 10, respectively. With regard to a rigorous error estimation, the AGP succeeds for the linear elastostatic problem but fails for the Helmholtz acoustic problem when an RB size is one at parameter values incurring only resonance. Otherwise, it manages to attain a rigorous error estimation. To conclude, the AGP is observed to be heuristically more practical than the SCM for a certified RB analysis. This research may considerably aid a static condensation RB element (scRBE) method because the technique numerously repeats the a posteriori error estimation of a reduced bubble function for the a posteriori error estimation of an scRBE solution.