A compressive Landweber iteration for solving ill-posed inverse problems
A compressive Landweber iteration for solving ill-posed inverse problems
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DOI：
10.1088/0266-5611/24/6/065013
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发表时间：
2008-12
期刊：
影响因子：
2.1
通讯作者：
R. Ramlau;Gerd Teschke;M. Zhariy
中科院分区：
文献类型：
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作者：
R. Ramlau;Gerd Teschke;M. Zhariy
在本文中，我们将考虑构造一个自适应的Landweber迭代来解线性不适定和反问题。经典的Landweber迭代格式与适当的正则化参数规则相结合，提供了排序最优的正则化格式。然而，对于许多应用，Landweber方法的实现在数值上是非常密集的。因此，我们提出了一种Landweber迭代的自适应变体，它可以显著减少计算开销，即导致Landweber迭代的压缩版本。我们借用了最初为适定算子方程(特别是椭圆型偏微分方程组)发展起来的自适应概念，本质上利用了小波(框架)、Besov正则性、最佳N项逼近的概念，并将其与经典的迭代正则化方案相结合。作为本文的主要结果，我们定义了Landweber迭代的一个自适应变量。结合适当的精化/停止规则(先验和后验原则)，我们证明了所提出的方法是一种正则化方法，对于精确和有噪声的数据是按范数收敛的。该方法在计算机层析成像领域得到了验证。
In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear ill-posed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber's method is numerically very intensive. Therefore we propose an adaptive variant of Landweber's iteration that may reduce the computational expense significantly, i.e. leading to a compressed version of Landweber's iteration. We borrow the concept of adaptivity that was primarily developed for well-posed operator equations (in particular, for elliptic PDE's) essentially exploiting the concept of wavelets (frames), Besov regularity, best N-term approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber's iteration. In combination with an adequate refinement/stopping rule (a priori as well as a posteriori principles) we prove that the proposed procedure is a regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.