Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse
Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse
复制标题
用近似逆方法非迭代求解Banach空间中的线性算子方程
DOI：
10.1088/0266-5611/26/8/085006
复制
发表时间：
2010
期刊：
影响因子：
2.1
通讯作者：
F. Schöpfer
中科院分区：
文献类型：
--
作者：
T. Schuster;F. Schöpfer
近似逆方法是稳定求解反问题的一种缓和方法。它的原始形式是为了求解 L2 空间和一般希尔伯特空间中的算子方程而开发的。我们证明了近似逆方法可以扩展到解决 Banach 空间中的线性不适定问题。本文仅限于功能空间。该方法本身包括对给定数据的对偶配对的评估以及与软化器和算子的对偶相关的重建内核。我们首先定义一般 Banach 空间中的缓和器的含义，然后更准确地研究两种设置：Lp 空间的情况和紧集上连续函数的 Banach 空间的情况。对于这两种设置，我们提出了将近似逆方法转变为正则化方法的标准，并证明了速率的收敛性。作为一种应用，我们指的是 X 射线衍射法，它是一种无损检测技术，涉及计算样本的应力张量。由于已知应力张量是平滑的，因此可以使用连续函数通过 Banach 空间设置适当地对 X 射线衍射进行建模。
The method of approximate inverse is a mollification method for stably solving inverse problems. In its original form it has been developed to solve operator equations in L2-spaces and general Hilbert spaces. We show that the method of approximate inverse can be extended to solve linear, ill-posed problems in Banach spaces. This paper is restricted to function spaces. The method itself consists of evaluations of dual pairings of the given data with reconstruction kernels that are associated with mollifiers and the dual of the operator. We first define what we mean by a mollifier in general Banach spaces and then investigate two settings more exactly: the case of Lp-spaces and the case of the Banach space of continuous functions on a compact set. For both settings we present the criteria turning the method of approximate inverse into a regularization method and prove convergence with rates. As an application we refer to x-ray diffractometry which is a technique of non-destructive testing that is concerned with computing the stress tensor of a specimen. Since one knows that the stress tensor is smooth, x-ray diffractometry can appropriately be modelled by a Banach space setting using continuous functions.
影响因子：
--
作者：
Bonesky, Thomas;Kazimierski, Kamil S.;Schuster, Thomas
通讯作者：
Schuster, Thomas