Boundary Measurements Research Articles

Imaging of permeability defect distribution by electromagnetic tomography with hybrid L1 normand nuclear normpenalty terms.

Electromagnetic tomography (EMT), with the advantages of being non-contact, non-invasiveness, low cost, simple structure, and fast imaging speed, is a multi-functional tomography technique based on boundary measurement voltages to image the conductivity distribution within the sensing field. EMT is widely used in industrial and biomedical fields. Currently, there are few studies on the application of EMT in magnetic permeability materials, which makes it difficult to obtain high-quality reconstructed images due to its own properties that lead to obvious attenuation of electromagnetic waves during propagation, as well as the ill-posed and ill-conditioned characteristics of EMT. In this paper, a multi-feature objective function integrating L2 normregularization, L1 normregularization, and low-rank normregularization is proposed to solve the challenge of magnetic permeability material imaging. This approach emphasizes the smoothness and sparsity. The split Bregman algorithm is introduced to efficiently solve the proposed objective function by decomposing the complex optimization problem into several simple sub-task iterative schemes. In addition, a nine-coil planar array electromagnetic sensor was developed and a flexible modular EMT system was constructed. We use correlation coefficient and error coefficient as indicators to evaluate the performance of the proposed image reconstruction algorithm. The effectiveness of the proposed method in improving the reconstruction accuracy and robustness is verified through numerical simulations and experiments.

Domain sampling methods for an inverse boundary value problem of the heat equation

Abstract We investigate the reconstruction of an unknown cavity inside a heat conductor with only single boundary measurement, which is a typical non-destructive testing of defects in material science using thermography. Mathematically, it can be formulated as an inverse boundary value problem for the heat equation. Two domain sampling methods, i.e., the range test (RT) and no-response test (NRT), are developed to realize the reconstruction. We first justify the convergence of the RT, which is based on the analytical extension property of solutions to the heat equation. Then we prove the duality of the RT and NRT by looking at the relation between their indicator functions. Consequently, the unknown cavity can be reconstructed by using either the RT or NRT if the solution associated to the single measurement cannot be analytically extended across its boundary. Next, we design two efficient algorithms for the RT and NRT to reconstruct the unknown cavity numerically. Each algorithm consists of two steps. The first step is to determine an approximation of the unknown cavity by implementing the sampling methods in a coarse grid, while the second step is to improve the reconstruction by using a fine grid in the approximate domain. The numerical results suggest that both algorithms can generate some rough information on the cavity with only one boundary measurement. Finally, by using this information as an initial guess for Newton's method, we improve the accuracy of the mentioned reconstruction results.

Uniqueness, stability and algorithm for an inverse wave-number-dependent source problem

Abstract This paper is concerned with an inverse wave-number-/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but independent on one spatial variable. The dependence of the source function on wave-number/frequency is supposed to be unknown. Based on the Fourier-transform and Dirichlet-Laplacian methods, we develop two efficient non-iterative numerical algorithms to recover the wave-number-dependent source. Uniqueness proof and increasing stability analysis are carried out if the boundary measurement data of Dirichlet kind are available. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed methods.

Adaptive fault estimation and accommodation for distributed parameter systems with coupled parabolic partial differential equations

This paper presents a novel model-based approach for fault detection, estimation and accommodation in linear distributed parameter systems (DPS) governed by parabolic partial differential equations (PDEs), prone to sensor and actuator faults. Unlike traditional methods, our approach utilises a filter-based observer directly employing the PDE representation and relies solely on boundary measurements, eliminating the need for extensive sensor arrays. By generating fault detection residuals from a comparison of measured and observer outputs, our method offers efficient fault detection. Innovative parameter update laws facilitate accurate estimation of fault parameters, enabling precise adjustment of the nominal controller to mitigate fault effects. Rigorous Lyapunov analysis ensures the robustness of our approach. Additionally, we derive a formula for estimating time-to-accommodation (TTA), providing valuable insights into fault resolution timelines. Simulations on a linearised diffusion process validate the effectiveness of our scheme, highlighting its superiority over existing approaches.

Analysis and computation of an inverse source problem for the biharmonic wave equation

Abstract In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on the Fourier series expansion by multi-wavenumber boundary measurements. The stability of the proposed method is analyzed and numerical experiments are presented to verify the accuracy and efficiency.

Determination of a small elliptical anomaly in electrical impedance tomography using minimal measurements

Abstract We consider the problem of determining a small elliptical conductivity anomaly in a unit disc from boundary measurements. The conductivity of the anomaly is assumed to be a small perturbation from the constant background. A measurement of voltage across two point-electrodes on the boundary through which a constant current is passed. We further assume the limiting case when the distance between two electrodes go to zero, creating a dipole field. We show that three such measurements suffice to locate the anomaly size and location inside the disc. Two further measurements are needed to obtain the aspect ratio and the orientation of the ellipse. The investigation includes the studies of the stability of the inverse problem and optimal experiment design.

Sensitivity Matrix Update Method for Electrical Resistance Tomography Based on Error-Constrained Cross Fusion Residual Attention Network

Abstract Electrical resistance tomography (ERT) is an imaging technique of conductivity distribution. The image reconstruction algorithm based on the empty field sensitivity matrix is widely used because it provides an approximate solution to the complex ERT inverse problems. However, due to the soft field effect, the sensitivity matrix changes with the media distribution, leading to significant errors in image reconstruction. To address this issue, an error-constrained deep learning scheme for bubble flow, namely cross fusion residual attention network (CFRA-Net) is designed to update the sensitivity matrix during gas-liquid-solid fluidized bed operation. CFRA-Net employs a multi-head self-attention mechanism to enhance bubble features in boundary measurements, a multilevel cross fusion module to fuse empty field strength and boundary measurement information, and a novel serial-channel residual spatial attention block to boost the feature extraction capability of the model. A weighted loss function is designed to give more weight to the gas phase distribution region. Additionally, this paper establishes a dataset for gas-liquid-solid three-phase flow, considering background field conductivity variations. Validation and reliability of the proposed method are assessed through ablation, comparison, and noise experiments. The experimental results demonstrate that CFRA-Net achieves rapid updates of the sensitivity matrix, high imaging accuracy, and robust noise immunity.

Quantifying the morphology of rock joints and updating the JRC–JCS criterion considering the asperity distribution

Roughness ubiquitously prevails in rock joints and controls the shear behaviours, permeability and damage characteristics of rock joints. A plethora of investigations have focused on the description of joint roughness; however, a detailed method for quantifying joint roughness and evaluating the shear strength has not yet been established. In this study, within the framework of fractal theory, an optical measurement scale was defined to depict the fractal characteristics of joint roughness, and a boundary measurement scale was used to identify first- and second-order asperities. A composite indicator η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document}, including the fractal roughness coefficient (RD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{D}$$\\end{document}), the coefficient describing the roughness order (μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document}) and the anisotropy parameter (λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda$$\\end{document}), was proposed to quantify the surface morphology, which takes the asperity distribution and roughness anisotropy into account. The relationship between η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document} and JRC was established, and the JRC–JCS criterion was further updated. Moreover, representative examples were given to show the steps required to quantify the morphology of rough joint surfaces with the new quantitative parameter. Direct shear tests were conducted to validate the effectiveness of the proposed method in describing joint roughness and estimating joint shear strength; the results indicate that it is appropriate to use η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta$$\\end{document} to estimate the joint roughness and that the proposed shear strength criterion can appropriately predict the shear strength within an acceptable error.

Observer-Based Asynchronous Boundary Stabilization for Stochastic Markovian Reaction-Diffusion Neural Networks.

This work investigates the observer-based asynchronous boundary stabilization for a kind of stochastic Markovian reaction-diffusion neural networks with exogenous disturbances. Specifically, parameter uncertainties are considered in the drift item. First, a hidden Markov model is introduced that guarantees the observer modes run asynchronously with the system modes. It should be noted that the asynchronous observer constructed in this work only uses the boundary measurement information. Then a nonfragile asynchronous observer-based boundary controller is designed. Taking advantage of inequality techniques and stochastic analysis method, sufficient criterion is provided to satisfy input-to-state exponentially mean-square stability, and the asynchronous boundary observer/controller gains are further derived. As a special case, the synchronous observer-based boundary stabilization is also obtained. Finally, a numerical example is exploited to manifest the validity of the established results.

Determining a time‐varying potential in time‐fractional diffusion from observation at a single point

AbstractWe discuss the identification of a time‐dependent potential in a time‐fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by leveraging the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.

Output feedback finite‐time boundary control for an unstable heat PDE with spatially varying coefficients

AbstractThis article studies the output feedback finite‐time boundary control for unstable heat systems with the spatially varying coefficient. First, a finite‐time observer with switched gains under a state‐dependent switching law is designed in order to estimate the states of the system in a finite‐time only exerting one displacement boundary measurement. Next, an observer‐based linear finite‐time control is planned. Namely, a linear switched control under a state‐dependent switching law is proposed to vanish every solution of an unstable heat partial differential equation with spatially varying coefficients in a finite time. We also present explicit forms for the proposed observer gains and output feedback finite‐time controller. Finally, some numerical simulations are provided to confirm the theoretical results.

Iterative Stress Reconstruction Algorithm to Estimate Three-Dimensional Residual Stress Fields in Manufactured Components

Abstract Residual stress (RS) significantly impacts the mechanical performance of components. Measurement of RS often provides incomplete data in terms of components of stress and spatial density. Employing such fields in finite element simulations results in significant modification of the field to achieve equilibrium and compatibility among strains. To overcome this, an iterative stress reconstruction algorithm (ISRA) is developed to estimate 3D RS fields that satisfy equilibrium, are stress component-wise complete, and represent the characterized data sampled. An Al 7075-T651 plate and an additively manufactured (AM) A36 steel wall are considered for RS reconstruction using measurement data from the literature. A maximum variation of ∼2.5 MPa in the Al plate, and ∼10 MPa in the steel wall are observed between the reconstructed and measured stresses. Furthermore, unknown stress components emerge and reach significant magnitudes (upto ∼2.3 MPa in the Al plate and ∼45 MPa in the AM wall) during ISRA. Indeed, it is found that minor errors in measurement or data processing are eliminated through the physical requirements during ISRA. Employing a reconstructed RS field is hence not just more accurate given its compatibility, but it additionally corrects for minor errors in measurement. Furthermore, it is found that spatially dense measurement data result in convergence with fewer iterations. Finally, although ISRA yields a nonunique solution dependent on boundary conditions, measurement errors, fitting errors, and mesh density, it accommodates for uncertainties and inaccuracies in measurement, as opposed to failing to reach a physically realistic converged solution.

Boundary measurement tomography of the Bose Hubbard model on general graphs

Correlated quantum many-body phenomena in lattice models have been identified as a set of physically interesting problems that cannot be solved classically. Analog quantum simulators, in photonics and microwave superconducting circuits, have emerged as near-term platforms to address these problems. An important ingredient in practical quantum simulation experiments is the tomography of the implemented Hamiltonians—while this can easily be performed if we have individual measurement access to each qubit in the simulator, this could be challenging to implement in many hardware platforms. In this paper, we present a scheme for tomography of quantum simulators which can be described by a Bose-Hubbard Hamiltonian while having measurement access to only some sites on the boundary of the lattice. We present an algorithm that uses the experimentally routine transmission and two-photon correlation functions, measured at the boundary, to extract the Hamiltonian parameters at the standard quantum limit. Furthermore, by building on quantum enhanced spectroscopy protocols that, we show that with the additional ability to switch on and off the on-site repulsion in the simulator, we can sense the Hamiltonian parameters beyond the standard quantum limit. Published by the American Physical Society 2024

A nonsmooth primal-dual method with interwoven PDE constraint solver

We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

Feasibility of acousto-electric tomography

In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.

Boundary containment control of multi-agent systems with time-invariant and time-varying delays

This paper considers the multi-leader situation in multi-agent systems (MASs), providing a solution to the issue of containment control in MASs using a directed graph as the main focus of the investigation. Containment controllers are first designed for MASs respectively by using boundary measurement and distributed measurement, as well as considering time-invariant and time-varying delays. By making use of the Wirtinger inequality and the Lyapunov Krasovskii function, consensus criteria for containment consensus of the MASs are established. Finally, two numerical examples are given to verify the effectiveness of the proposed protocols.

Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurement

In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper a priori information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms.

Simultaneous uniqueness for the diffusion coefficient and initial value identification in a time-fractional diffusion equation

This article investigates the uniqueness of simultaneously determining the diffusion coefficient and initial value in a time-fractional diffusion equation with derivative order α∈(0,1). By additional boundary measurements and a priori assumption on the diffusion coefficient, the uniqueness of the eigenvalues and an associated integral equation for the diffusion coefficient are firstly established. The proof is based on the Laplace transform and the expansion of eigenfunctions for the solution to the initial value/boundary value problem. Furthermore, by using these two results, the simultaneous uniqueness in determining the diffusion coefficient and initial value is demonstrated from the Liouville transform and Gelfand–Levitan theory. The result shows that the uniqueness in simultaneous identification can be achieved, provided the initial values non-orthogonality to the eigenfunction of differential operators, which incorporates only one diffusion coefficient rather than scenarios involving two diffusion coefficients.

Inverse Source Problem in Radiative Transport with Nonlinear Anisotropic Sources

An inverse source problem in the stationary radiative transport in two dimension in an absorbing and scattering medium is considered, where the source is directionally quadratic polynomial. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. The attenuating and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier coefficients in the angular variable, we show how to recover the nonlinear anisotropic radiative source from boundary measurements.

Finite-Dimensional Boundary Control of a Wave Equation With Viscous Friction and Boundary Measurements

Finite-Dimensional Boundary Control of a Wave Equation With Viscous Friction and Boundary Measurements