Bridging and Improving Theoretical and Computational Electrical Impedance Tomography via Data Completion
Bridging and Improving Theoretical and Computational Electrical Impedance Tomography via Data Completion
复制标题
通过数据补全桥接和改进理论和计算电阻抗断层扫描
DOI：
10.1137/21m141703x
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发表时间：
2022
影响因子：
3.1
通讯作者：
Zepeda-Nún͂ez, Leonardo
中科院分区：
文献类型：
--
作者：
Bui-Thanh, Tan;Li, Qin;Zepeda-Nún͂ez, Leonardo
在基于计算偏微分方程的反问题中，收集有限数量的数据来推断偏微分方程中的未知参数。为了获得准确的推断，收集的数据必须是关于未知参数的信息。如何确定哪些数据是最具信息性的，以及如何有效地对其进行采样，是最优实验设计（OED）中众所周知的具有挑战性的任务。在这种情况下，最好的，而且往往是不可行的情况是，当完整的输入到输出（ItO）映射，即，无限数量的数据是可用的：这是许多理论逆问题中的典型设置，用于保证唯一的参数重建。这两种不同的设置在计算和理论逆问题之间产生了差距，其中分别使用有限和无限数量的数据。在这篇文章中，我们的目标是弥合这一差距，同时绕过牛津英语词典的任务。这是通过利用的ItO数据的结构，从底层的逆问题，使用电阻抗断层扫描（EIT）问题作为一个例子。为了实现我们的目标，我们利用EIT模型的秩结构，并将ItO矩阵\textmdash离散化ItO映射\textmdash表示为非对角块秩较低的矩阵。这表明，当配备有矩阵完成技术时，可以从遵循秩结构采样的其条目的子集以高概率恢复完整的ItO矩阵：对角块中的数据是信息性的并且应该被完全采样，而非对角块中的数据可以被二次采样。然后，利用恢复的ItO矩阵来呈现完整的ItO图，直到离散化误差，为在保证参数的唯一重构的理论设置中与问题连接铺平了道路。这种策略实现了两个目标：（I）它的桥梁之间的差距差距有限维和无限维的数值和理论的逆问题的设置和（II）它提高了计算逆解的质量。我们详细的EIT模型的理论，并提供数值验证EIT和光学层析成像问题。
In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data must be informative about the unknown parameters. How to decide which data is most informative and how to efficiently sample it is the notoriously challenging task of optimal experimental design (OED). In this context, the best, and often infeasible, scenario is when the full input-to-output (ItO) map, i.e., an infinite amount of data, is available: This is the typical setting in many theoretical inverse problems, which is used to guarantee the unique parameter reconstruction. These two different settings have created a gap between computational and theoretical inverse problems, where finite and infinite amounts of data are used, respectively. In this article we aim to bridge this gap while circumventing the OED task. This is achieved by exploiting the structures of the ItO data from the underlying inverse problem, using the electrical impedance tomography (EIT) problem as an example. To accomplish our goal, we leverage the rank structure of the EIT model and formulate the ItO matrix\textemdash the discretized ItO map\textemdash as an-matrix whose off-diagonal blocks are low rank. This suggests that, when equipped with the matrix completion technique, one can recover the full ItO matrix, with high probability, from a subset of its entries sampled following the rank structure: The data in the diagonal blocks is informative and should be fully sampled, while data in the off-diagonal blocks can be subsampled. This recovered ItO matrix is then utilized to present the full ItO map up to a discretization error, paving the way to connect with the problem in the theoretical setting where the unique reconstruction of parameters is guaranteed. This strategy achieves two goals: (I)it bridges the gap between the finite- and infinite-dimensional settings for numerical and theoretical inverse problemsand (II)it improves the quality of computational inverse solutions. We detail the theory for the EIT model and provide numerical verification to both EIT and optical tomography problems.