ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD FOR NONLINEAR INVERSE PROBLEMS WITH RANDOM NOISE
ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD FOR NONLINEAR INVERSE PROBLEMS WITH RANDOM NOISE
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DOI：
10.1137/080721789
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发表时间：
2009-01-01
影响因子：
2.9
通讯作者：
Munk, Axel
中科院分区：
文献类型：
--
作者：
Bauer, Frank;Hohage, Thorsten;Munk, Axel
研究了Hilbert空间中正则化Newton方法应用于非线性算子方程的收敛性。它表明，预期的平方误差是有界的一个常数乘以相应的线性化问题的极大极小率，如果停止指标选择使用先验知识的解决方案的平滑。对于未知的光滑性，停止指标可以根据Lepskii平衡原理自适应地选择。对于这个停止规则，我们建立了一个预言不等式，这意味着为了确定性错误的最优速率，和最优速率的对数因子的随机噪声。所提出的方法的性能和统计特性说明了Monte Carlo模拟。
We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. It is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution. For unknown smoothness the stopping index can be chosen adaptively based on Lepskii's balancing principle. For this stopping rule we establish an oracle inequality, which implies order optimal rates for deterministic errors, and optimal rates up to a logarithmic factor for random noise. The performance and the statistical properties of the proposed method are illustrated by Monte Carlo simulations.