{"title":"An analysis of discontinuous Galerkin method for Electrical Impedance Tomography with partial data","authors":"Xiaosheng Li, Wei Wang","doi":"10.1016/j.cam.2024.116376","DOIUrl":null,"url":null,"abstract":"<div><div>In this work, we extend our previous work (Li and Wang, 2023) of the discontinuous Galerkin (DG) method for Electrical Impedance Tomography (EIT) with full data to partial data where the current and voltage measurements are taken only on part of the boundary. Additionally, we provide the convergence analysis of the DG approximation for EIT for partial data based on an iterative method with Tikhonov regularization. We prove that the minimizers of the discrete optimization problems converge to a minimizer of the continuous optimization problem as the mesh sizes of the discretization approach zero. Numerical results for the recovery of conductivities are tested with different types of partial data. The partial data results are demonstrated to be comparable to the full data results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116376"},"PeriodicalIF":2.1000,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724006241","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we extend our previous work (Li and Wang, 2023) of the discontinuous Galerkin (DG) method for Electrical Impedance Tomography (EIT) with full data to partial data where the current and voltage measurements are taken only on part of the boundary. Additionally, we provide the convergence analysis of the DG approximation for EIT for partial data based on an iterative method with Tikhonov regularization. We prove that the minimizers of the discrete optimization problems converge to a minimizer of the continuous optimization problem as the mesh sizes of the discretization approach zero. Numerical results for the recovery of conductivities are tested with different types of partial data. The partial data results are demonstrated to be comparable to the full data results.
在这项工作中，我们将之前的工作（Li and Wang, 2023）中用于电阻抗断层扫描（EIT）的非连续伽勒金（DG）方法（全数据）扩展到了部分数据，即仅在部分边界上进行电流和电压测量。此外，我们还提供了基于 Tikhonov 正则化迭代法的部分数据 EIT DG 近似的收敛性分析。我们证明，当离散化的网格尺寸趋近于零时，离散优化问题的最小值会收敛到连续优化问题的最小值。利用不同类型的部分数据对恢复电导率的数值结果进行了测试。结果表明，部分数据结果可与完整数据结果相媲美。
期刊介绍：
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.
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