Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
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DOI：
10.1007/s00211-012-0499-z
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发表时间：
2013-04-01
影响因子：
2.1
通讯作者：
Werner, Frank
中科院分区：
文献类型：
--
作者：
Hohage, Thorsten;Werner, Frank
我们研究巴纳赫空间中非线性算子方程描述的反问题的牛顿型方法，其中牛顿方程使用一般数据失配函数和凸正则化项进行变分正则化。这概括了著名的迭代正则化高斯-牛顿方法 (IRGNM)。我们证明了先验停止规则和 LepskiA 型后验停止规则的噪声水平趋向于收敛和收敛速度。我们的分析包括 IRGNM 的前阶最优收敛率结果（作为特殊情况）。本文的主要重点是泊松数据的反演问题，其中自然数据失配函数由 Kullback-Leibler 散度给出。详细讨论了此类问题的两个例子：具有远场方向图幅度数据的逆障碍物散射问题和相位检索问题。数值示例说明了所提出的方法针对这些问题的性能。
We study Newton type methods for inverse problems described by nonlinear operator equations in Banach spaces where the Newton equations are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to both for an a priori stopping rule and for a LepskiA-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performance of the proposed method for these problems is illustrated in numerical examples.