Local ill-posedness and source conditions of operator equations in Hilbert spaces
Local ill-posedness and source conditions of operator equations in Hilbert spaces
复制标题
希尔伯特空间算子方程的局部不适定性和源条件
DOI：
10.1088/0266-5611/14/5/007
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
O. Scherzer
中科院分区：
文献类型：
--
作者：
Bernd Hofmann;O. Scherzer
局部不适定性和局部非线性度的刻画对于非线性不适定问题的稳定解具有特别重要的意义。我们提出了关于非线性问题的不适定性与其线性化之间的相互关系的断言。此外，我们还证明了非线性程度与源条件相结合的概念可以用来刻画非线性不适定问题的局部不适定性，并推导出非线性不适定问题的后验估计。后验估计在有限元和多重网格法中被广泛应用于求解非线性偏微分方程组，但这些技术一般不适用于反问题和不适定问题。此外，我们还证明了对于著名的Landweber方法和迭代正则化的Gauss-牛顿方法，它们在源条件下满足后验估计；这可以用来证明收敛速度结果。给出了一个数值试验，证实了理论断言。
The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interpendence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity combined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse and ill-posed problems. Additionally we show for the well known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rate results. A numerical test is presented that confirms the theoretical assertions.