本文致力于证明变分和迭代正则化方法在变分源条件 VSCs 下的收敛速率，这些方法适用于其线性化满足范围不变性条件的反问题。为了实现这一目标，通常需要找到问题的适当松弛，这种松弛通常基于未知数集合的扩充，并导致反问题的特殊结构化重构。我们分析了利用这种结构的三种方法，即变分和 Newton 型方案，其无速率的收敛性已在 \cite{rangeinvar} 中建立；此外，我们提出了一种分裂最小化方法，并证明其满足相同的速率结果。范围不变性条件已在多个偏微分方程系数识别问题中得到验证，这些问题基于边界观测，与多种断层成像模态相关。我们的动机特别来自于电阻抗断层成像 EIT 这一经典反问题，我们研究了其通过扩散型方程的原始表述及其作为 Schrödinger 方程的重构。对于这两种表述，我们找到了可证明满足范围不变性条件的松弛。结合 \cite{Diss-Weidling} 中关于 VSCs 的结果与上述三种方法的抽象框架，我们得到了 EIT 中变分、分裂最小化和 Newton 型方法的收敛速率结果。
英文摘要：
This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \\ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schrödinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT.
This work considers electrical impedance tomography imaging of the human head, with the ultimate goal of locating and classifying a stroke in emergency care. One of the main difficulties in the envisioned application is that the electrode locations and the shape of the head are not precisely known, leading to significant imaging artifacts due to impedance tomography being sensitive to modeling errors. In this study, the natural variations in the geometry of the head and skull are modeled based on a library of head anatomies. The effect of these variations, as well as that of misplaced electrod…
Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The idea is to not define the electrodes as strict geometric objects on the measurement boundary, but only to assume approximate knowledge about their whereabouts and let a boundary admittivity function determine the actual locations of the current inputs…
An Analysis of Finite Element Approximation in Electrical Impedance Tomography
Center for Industrial Mathematics, University of Bremen（不来梅大学工业数学中心）；Department of Mathematics, University of California, Riverside（加州大学河滨分校数学系）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
This work investigates the electrical impedance tomography (EIT) problem when only limited boundary measurements are available, which is known to be challenging due to the extreme ill-posedness. Based on the direct sampling method (DSM), we propose deep direct sampling methods (DDSMs) to locate inhomogeneous inclusions in which two types of deep neural networks (DNNs) are constructed to approximate the index function(functional): fully connected neural network(FNN) and convolutional neural network (CNN). The proposed DDSMs are easy to be implemented, capable of incorporating multiple Cauchy da…
A path integral Monte Carlo method (PIMC) based on Feynman-Kac formula for mixed boundary conditions of elliptic equations is proposed to solve the forward problem of electrical impedance tomography (EIT) on the boundary to obtain electrical potentials. The forward problem is an important part for iterative algorithms of the inverse problem of EIT, which has attracted continual interest due to its applications in medical imaging and material testing of materials. By simulating reflecting Brownian motion with walk-on-sphere techniques and calculating its corresponding local time, we are able to…
Determining physical properties inside an object without access to direct measurements of target regions can be formulated as a specific type of \textit{inverse problem}. One of such problems is applied in \textit{Electrical Impedance Tomography} (EIT). In general, EIT can be posed as a minimization problem and solved by iterative methods, which require knowledge of derivatives of the objective function. In practice, this can be challenging because analytical closed-form solutions for them are hard to derive and implement efficiently. In this paper, we study the effectiveness of \textit{automa…
As second-order methods, Gauss--Newton-type methods can be more effective than first-order methods for the solution of nonsmooth optimization problems with expensive-to-evaluate smooth components. Such methods, however, often do not converge. Motivated by nonlinear inverse problems with nonsmooth regularization, we propose a new Gauss--Newton-type method with inexact relaxed steps. We prove that the method converges to a set of disjoint critical points given that the linearisation of the forward operator for the inverse problem is sufficiently precise. We extensively evaluate the performance o…
This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical result…
This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_{2}$). A general form of the Fréchet gradient is systematically derived by optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on $\mathbb{S}^{1}$ is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to $O(N)$ from $O(N^{3})$ of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach fo…
This paper introduces a constructive method for approximating relative continuum measurements in two-dimensional electrical impedance tomography based on data originating from either the point electrode model or the complete electrode model. The upper bounds for the corresponding approximation errors explicitly depend on the number (and size) of the employed electrodes as well as on the regularity of the continuum current that is mimicked. In particular, if the input current and the object boundary are infinitely smooth, the discrepancy associated with the point electrode model converges to ze…
Objective: To present the first 3D CGO-based absolute EIT reconstructions from experimental tank data. Approach: CGO-based methods for absolute EIT imaging are compared to traditional TV regularized non-linear least squares reconstruction methods. Additional robustness testing is performed by considering incorrect model\textbf{}ing of domain shape. Main Results: The CGO-based methods are fast, and show strong robustness to incorrect domain modeling comparable to classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear least squares reference reconstruction…
