Goal oriented adaptivity in the IRGNM for parameter identification in PDEs: II. all-at-once formulations
Goal oriented adaptivity in the IRGNM for parameter identification in PDEs: II. all-at-once formulations
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DOI：
10.1088/0266-5611/30/4/045002
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发表时间：
2014-04-01
期刊：
影响因子：
2.1
通讯作者：
Vexler, B.
中科院分区：
文献类型：
--
作者：
Kaltenbacher, B.;Kirchner, A.;Vexler, B.
本文研究了迭代正则化高斯-牛顿方法IRGNM的自适应离散化。与Kaltenbacher等人（2014 Inverse Problems 30 045001）的简化形式相比，同时考虑PDE和测量方程的一次性公式允许避免在每个牛顿步骤中潜在非线性PDE的（近似）解。我们分析了最小二乘法和广义高斯-牛顿公式，并在这两种情况下证明收敛和收敛速度与后验选择的正则化参数在每个牛顿步骤和停止指数在一定的精度要求下的四个数量的兴趣。通过加权对偶残差法估计这些量的误差进行了讨论，这导致了自适应网格细化算法。数值实验表明，该算法的数值效率，特别是对于强非线性偏微分方程，优于Kaltenbacher等人所考虑的非线性Tikhonov正则化
In this paper we investigate adaptive discretization of the iteratively regularized Gauss-Newton method IRGNM. All-at-once formulations considering the PDE and the measurement equation simultaneously allow to avoid (approximate) solution of a potentially nonlinear PDE in each Newton step as compared to the reduced form Kaltenbacher et al (2014 Inverse Problems 30 045001). We analyze a least squares and a generalized Gauss-Newton formulation and in both cases prove convergence and convergence rates with a posteriori choice of the regularization parameters in each Newton step and of the stopping index under certain accuracy requirements on four quantities of interest. Estimation of the error in these quantities by means of a weighted dual residual method is discussed, which leads to an algorithm for adaptive mesh refinement. Numerical experiments with an implementation of this algorithm show the numerical efficiency of this approach, which especially for strongly nonlinear PDEs outperforms the nonlinear Tikhonov regularization considered in Kaltenbacher et al