SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1450-1471, June 2022.
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.

中文翻译：

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椭圆线性二次参数估计问题中参数误差的自适应 FEM

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1450-1471 页，2022 年 6 月。
我们考虑具有有限数量参数的椭圆线性二次参数估计问题。证明了一种新的参数误差先验界，并基于该界，提出了一种由后验误差估计器驱动的自适应有限元方法。与文献中的先前结果不同，我们的估计器由状态方程的标准能量误差残差估计器和合适的共态问题组成，与（共）态变量相比，反映了参数误差的更快收敛。我们展示了我们方法的最佳收敛速度；特别是，与以前的工作不同，我们证明了估计量的下降率是状态和共态变量的最佳近似率之和。实验证实我们的方法与参数误差的收敛速度相匹配。