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Nonlinear solution approaches for inverse problems require fast simulation techniques for the underlying sensing problem. In this work, the authors investigate finite element (FE) based sensor simulations for the inverse problem of electrical capacitance tomography. Two known computational bottlenecks are the assembly of the FE equation system as well as the computation of the Jacobian. Here, existing computation techniques like adjoint field approaches require additional simulations. This paper aims to present fast numerical techniques for the sensor simulation and computations with the Jacobian matrix.

For the FE equation system, a solution strategy based on Green’s functions is derived. Its relation to the solution of a standard FE formulation is discussed. A fast stiffness matrix assembly based on an eigenvector decomposition is shown. Based on the properties of the Green’s functions, Jacobian operations are derived, which allow the computation of matrix vector products with the Jacobian for free, i.e. no additional solves are required. This is demonstrated by a Broyden–Fletcher–Goldfarb–Shanno-based image reconstruction algorithm.

MATLAB-based time measurements of the new methods show a significant acceleration for all calculation steps compared to reference implementations with standard methods. E.g. for the Jacobian operations, improvement factors of well over 100 could be found.

The paper shows new methods for solving known computational tasks for solving inverse problems. A particular advantage is the coherent derivation and elaboration of the results. The approaches can also be applicable to other inverse problems.

Within the framework of image reconstruction in cylindrical electrical capacitance tomography (ECT) sensors, the purpose of this study is to select the structure of a sensor in terms of number and size of the electrodes, to predict the radius and the position of a single circular shape lying in the cross-section defined by the sensor electrodes.

Nonlinear black-box models using a set of physically independent capacitances and least-square support vector machines models selected with a sophisticated validation method are implemented.

The coordinates of circular shapes are well estimated in fixed and variable permittivity environments even with noisy data. Various numerical experiments are presented and discussed. Sensors formed by three or four electrodes covering 50 per cent of the sensor perimeter provide the best prediction performances.

The proposed method is limited to the detection of a single circular shape in a cylindrical ECT sensor.

This method can be advantageously implemented in real-time applications, as it is numerically cost-effective and necessitates a small amount of measurements.

The contribution is two-fold: a fast computation of a circular shape position and radius with a satisfactory precision compared to the sensor size, and the determination of a cylindrical ECT sensor architecture that allows the most efficient predictions.

Inverse problems are often marked by highly dimensional state vectors. The high dimension affects the quality of the estimation result as well as the computational complexity of the estimation problem. This paper aims to present a state reduction technique based on prior knowledge.

Ill-posed inverse problems require prior knowledge to find a stable solution. The prior distribution is constructed for the high-dimensional data space. The authors use the prior distribution to construct a reduced state description based on a lower-dimensional basis, which they derive from the prior distribution. The approach is tested for the inverse problem of electrical capacitance tomography.

Based on a singular value decomposition of a sample-based prior distribution, a reduced state model can be constructed, which is based on principal components of the prior distribution. The approximation error of the reduced basis is evaluated, showing good behavior with respect to the achievable data reduction. Owing to the structure, the reduced state representation can be applied within existing algorithms.

The full state description is a linear function of the reduced state description. The reduced basis can be used within any existing reconstruction algorithm. Increased noise robustness has been found for the application of the reduced state description in a back projection-type reconstruction algorithm.

The paper presents the construction of a prior-based state reduction technique. Several applications of the reduced state description are discussed, reaching from the use in deterministic reconstruction methods up to proposal generation for computational Bayesian inference, e.g. Markov chain Monte Carlo techniques.

The purpose of this paper is to view the flow concentration of the flowing material in a pipeline conveyor.

Optical tomography provides a method to view the cross sectional image of flowing materials in a pipeline conveyor. Important flow information such as flow concentration profile, flow velocity and mass flow rate can be obtained without the need to invade the process vessel. The utilization of powerful computer together with expensive data acquisition system (DAQ) as the processing device in optical tomography systems has always been a norm. However, the advancements in silicon fabrication technology nowadays allow the fabrication of powerful digital signal processors (DSP) at reasonable cost. This allows the technology to be applied in optical tomography system to reduce or even eliminate the need of personal computer and the DAQ. The DSP system was customized to control the data acquisition of 16 × 16 optical sensors (arranged in orthogonal projection) and 23 × 23 optical sensors (arranged in rectilinear projections). The data collected were used to reconstruct the cross sectional image of flowing materials inside the pipeline. In the developed system, the accuracy of the image reconstruction was increased by 12.5 per cent by using new hybrid image reconstruction algorithm.

The results proved that the data acquisition and image reconstruction algorithm is capable of acquiring accurate data to reconstruct cross sectional images with only little error compared to the expected measurements.

The DSP system was customized to control the data acquisition of 16 × 16 optical sensors (arranged in orthogonal projection) and 23 × 23 optical sensors (arranged in rectilinear projections).

The purpose of this paper is to develop a program based on three‐dimensional finite element analysis to model different patterns of capacitive proximity sensors. This program can be used as a development tool to optimize the structure and size of a sensor for a desired or for a given sensitivity and linearity range and as a consequence to save sensor design time. A set of experiments have been conducted to test the tool capabilities for designing different sensor structures.

Finite element analysis in ANSYS software was used to perform electrostatic field simulations and to calculate the capacitance between electrodes of a capacitive proximity sensor when a conducting target is placed in some distance from the sensor plate.

Several capacitive proximity sensor structures have been designed, analyzed and tested to illustrate the accuracy of the simulated results obtained from the design tool. After design and implementation of a sensor and comparing the extracted and measured capacitance values, it is shown that the finite element analysis is an accurate method to extract fringing capacitance in capacitive proximity sensors in comparison to the analytical tool based on the finite difference method.

This automatic capacitive proximity sensor design tool can optimize a sensor structure with specific shape and size to have more sensitivity or linearity according to the application in use. Moreover, the modeling program can extract characteristics of a sensor with user‐defined parameters.

The application of statistical inversion theory provides a powerful approach for solving estimation problems including the ability for uncertainty quantification (UQ) by means of Markov chain Monte Carlo (MCMC) methods and Monte Carlo integration. This paper aims to analyze the application of a state reduction technique within different MCMC techniques to improve the computational efficiency and the tuning process of these algorithms.

A reduced state representation is constructed from a general prior distribution. For sampling the Metropolis Hastings (MH) Algorithm and the Gibbs sampler are used. Efficient proposal generation techniques and techniques for conditional sampling are proposed and evaluated for an exemplary inverse problem.

For the MH-algorithm, high acceptance rates can be obtained with a simple proposal kernel. For the Gibbs sampler, an efficient technique for conditional sampling was found. The state reduction scheme stabilizes the ill-posed inverse problem, allowing a solution without a dedicated prior distribution. The state reduction is suitable to represent general material distributions.

The state reduction scheme and the MCMC techniques can be applied in different imaging problems. The stabilizing nature of the state reduction improves the solution of ill-posed problems. The tuning of the MCMC methods is simplified.

The paper presents a method to improve the solution process of inverse problems within the Bayesian framework. The stabilization of the inverse problem due to the state reduction improves the solution. The approach simplifies the tuning of MCMC methods.

The purpose of this study is to develop a reconstruction and measurement system for data analysis using ultrasonic transmission tomography. The problem of reconstruction from the projection is encountered in practical implementation, which consists in reconstructing an image that is an estimation of an unknown object from a finite set of projection data. Reconstructive algorithms used in transmission tomography are based on linear mathematical models, which makes it necessary to process non-linear data into estimates for a finite number of projections. The application of transformation methods requires building a mathematical model in which the projection data forming the known and unknown quantities are functions with arguments from a continuous set of real numbers, determining the function describing the unknown quantities sought in the form of inverse relation and adapting it to operate on discrete and noisy data. This was done by designing a tomographic device and proprietary algorithms capable of reconstructing two-dimensional images regardless of the size, shape, location or number of inclusions hidden in the examined object.

The application consists of a device and measuring sensors, as well as proprietary algorithms for image reconstruction. Ultrasonic transmission tomography makes it possible to analyse processes occurring in an object without interfering with the examined object. The proposed solution uses algorithms based on ray integration, the Fermat principle and deterministic methods. Two applications were developed, one based on C and implemented on the embedded device, while the other application was made in Matlab.

Research shows that ultrasonic transmission tomography provides an effective analysis of tested objects in closed tanks.

In the presented technique, the use of ultrasonic absorption wave has been limited. Nevertheless, the effectiveness of such a solution has been confirmed.

The presented solution can be used for research and monitoring of technological processes.

Author’s tomographic system consisting of a measuring system and image reconstruction algorithms.

This study aims to design and validate a theoretical model for capacitive imaging (CI) sensors that incorporates the interelectrode shielding and surrounding shielding electrodes. Through experimental verification, the effectiveness of the theoretical model in evaluating CI sensors equipped with shielding electrodes has been demonstrated.

The study begins by incorporating the interelectrode shielding and surrounding shielding electrodes of CI sensors into the theoretical model. A method for deriving the semianalytical model is proposed, using the renormalization group method and physical model. Based on random geometric parameters of CI sensors, capacitance values are calculated using both simulation models and theoretical models. Three different types of CI sensors with varying geometric parameters are designed and manufactured for experimental testing.

The study’s results indicate that the errors of the semianalytical model for the CI sensor are predominantly below 5%, with all errors falling below 10%. This suggests that the semianalytical model, derived using the renormalization group method, effectively evaluates CI sensors equipped with shielding electrodes. The experimental results demonstrate the efficacy of the theoretical model in accurately predicting the capacitance values of the CI sensors.

The theoretical model of CI sensors is described by incorporating the interelectrode shielding and surrounding shielding electrodes into the model. This comprehensive approach allows for a more accurate evaluation of the detecting capability of CI sensors, as well as optimization of their performance.

This paper aims to present an approach based on electrical impedance tomography spectroscopy (EITS) for the determination of water and ice fraction in low-power applications such as autarkic wireless sensors, which require a low computational complexity reconstruction approach and a low number of electrodes. This paper also investigates how the electrode design can affect the reconstruction results in tomography.

EITS is performed by using a non-iterative method called optimal first order approximation. In addition to that, a planar electrode geometry is used instead of the traditional circular electrode geometry. Such a structure allows the system to identify materials placed on the region above the sensor, which do not need to be confined in a pipe. For the optimization, the mean squared error (MSE) between the reference images and the obtained reconstructed images was calculated.

The authors demonstrate that even with a low number of four electrodes and a low complexity reconstruction algorithm, a reasonable reconstruction of water and ice fractions is possible. Furthermore, it is shown that an optimal distribution of the sensor electrodes can help to reduce the MSE without any costs in terms of computational complexity or power consumption.

This paper shows through simulations that the reconstruction of ice and water mixtures is possible and that the electrode design is a topic of great importance, as they can significantly affect the reconstruction results.

The purpose of this paper is to explore gas/liquid two-phase flow is widely existed in industrial fields, especially in chemical engineering. Electrical resistance tomography (ERT) is considered to be one of the most promising techniques to monitor the transient flow process because of its advantages such as fast respond speed and cross-section imaging. However, maintaining high resolution in space together with low cost is still challenging for two-phase flow imaging because of the ill-conditioning of ERT inverse problem.

In this paper, a sparse reconstruction (SR) method based on the learned dictionary has been proposed for ERT, to accurately monitor the transient flow process of gas/liquid two-phase flow in a pipeline. The high-level representation of the conductivity distributions for typical flow regimes can be extracted based on denoising the deep extreme learning machine (DDELM) model, which is used as prior information for dictionary learning.

The results from simulation and dynamic experiments indicate that the proposed algorithm efficiently improves the quality of reconstructed images as compared to some typical algorithms such as Landweber and SR-discrete fourier transformation/discrete cosine transformation. Furthermore, the SR-DDELM has also used to estimate the important parameters of the chemical process, a case in point is the volume flow rate. Therefore, the SR-DDELM is considered an ideal candidate for online monitor the gas/liquid two-phase flow.

This paper fulfills a novel approach to effectively monitor the gas/liquid two-phase flow in pipelines. One deep learning model and one adaptive dictionary are trained via the same prior conductivity, respectively. The model is used to extract high-level representation. The dictionary is used to represent the features of the flow process. SR and extraction of high-level representation are performed iteratively. The new method can obviously improve the monitoring accuracy and save calculation time.