Sharp converse results for the regularization error using distance functions
Sharp converse results for the regularization error using distance functions
复制标题
使用距离函数的正则化误差的尖锐逆结果
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
P. Mathé
中科院分区：
文献类型：
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作者：
Jens Flemming;B. Hofmann;P. Mathé
在不适定反问题的分析中，解的平滑度对精度和收敛速度的影响起着重要作用。对于希尔伯特空间中的线性不适定算子方程，并重点关注线性正则化模式，我们将以逐点或积分的方式建立不同类型的测量解平滑度之间的关系。特别是，我们讨论了分布函数、表示正则化误差的轮廓函数、生成源条件的索引函数以及与基准源条件相关的距离函数的相互作用。我们表明，通常距离函数和轮廓函数携带与分布函数相同的信息，并且对于一般源条件而言情况并非如此。理论研究结果附有展示该方法的应用和局限性的示例。对解光滑度的详细理解也将有助于非线性不适定问题的处理和收敛分析。
In the analysis of ill-posed inverse problems the impact of solution smoothness on accuracy and convergence rates plays an important role. For linear ill-posed operator equations in Hilbert spaces and with focus on the linear regularization schema we will establish relations between different kinds of measuring solution smoothness in a point-wise or integral manner. In particular, we discuss the interplay of distribution functions, profile functions that express the regularization error, index functions generating source conditions and distance functions associated with benchmark source conditions. We show that typically the distance functions and the profile functions carry the same information as the distribution functions, and that this is not the case for general source conditions. The theoretical findings are accompanied with examples exhibiting applications and limitations of the approach. A detailed understanding of solution smoothness will also be helpful for the treatment and convergence analysis of nonlinear ill-posed problems.