2021/11/05 by Roland Becker, Michael Innerberger, Dirk Praetorius · 1 voice
Engineering · Mathematics · Physics and Astronomy · #A priori and a posteriori #Adaptive estimator #Advanced Numerical Methods in Computational Mathematics #Algorithm #Applied mathematics #Computer science #Convergence (economics) #Estimation theory #Estimator #Finite element method #Mathematical analysis #Mathematical optimization #Mathematics #Model Reduction and Neural Networks #Numerical methods in engineering #Quadratic equation #Rate of convergence #Residual #Statistics #Upper and lower bounds #math.NA #math.OC
paper · pdf · doi:10.1137/21m1458077
arxiv published 2021/11/05 · openalex created_date 2021/11/22 · arxiv updated 2022/02/07 · openalex publication_date 2022/06/01 · openalex updated_date 2026/08/01
We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature, our estimator, which is composed of standard energy error residual estimators for the state equation and suitable co-state problems, reflects the faster convergence of the parameter error compared to the (co-)state variables. We show optimal convergence rates of our method; in particular and unlike prior works, we prove that the estimator decreases with a rate that is the sum of the best approximation rates of the state and co-state variables. Experiments confirm that our method matches the convergence rate of the parameter error.