Landweber Iteration Research Articles

The inverse problem of identifying complex hyperbolic equation source terms in electromagnetic propagation

Abstract This article investigates the inverse problem of determining the source term of the hyperbolic equation for electromagnetic propagation using terminal data. This study is an important method for identifying propagation sources in electromagnetics. Unlike wave equations, the complexity of the underlying equations can make theoretical analysis quite difficult. Firstly, the uniqueness of the inverse problem was proved using the energy method. Then, based on the optimal control framework, the inverse problem was transformed into an optimal control problem, and the existence of the optimal solution and its necessary conditions were established. Secondly, the global uniqueness and stability of the optimal solution have been proven, which is a completely new conclusion. This has laid a solid theoretical foundation for numerical algorithms. Finally, it is proposed to apply the Landweber iteration method and conjugate gradient method to this problem, and some numerical examples are provided to demonstrate the effectiveness and convergence speed of these two algorithms.

Real-Time Nonlinear Image Reconstruction in Electrical Capacitance Tomography Using the Generative Adversarial Network

This study investigated the potential of the generative adversarial neural network (cGAN) image reconstruction in industrial electrical capacitance tomography. The image reconstruction quality was examined using image patterns typical for a two-phase flow. The training dataset was prepared by generating images of random test objects and simulating the corresponding capacitance measurements. Numerical simulations were performed using the ECTsim toolkit for MATLAB. A cylindrical sixteen-electrode ECT sensor was used in the experiments. Real measurements were obtained using the EVT4 data acquisition system. The reconstructed images were evaluated using selected image quality metrics. The results obtained using cGAN are better than those obtained using the Landweber iteration and simplified Levenberg–Marquardt algorithm. The suggested method offers a promising solution for a fast reconstruction algorithm suitable for real-time monitoring and the control of a two-phase flow using ECT.

Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces

In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.

A modified capacitance tomography image reconstruction approach based on iterative shrinkage-thresholding algorithm combined with deep networks

Abstract Due to the ‘soft-field’ effect and the challenges posed by ill-posed and ill-conditioned inverse problems, it is difficult to obtain high quality images from an electrical capacitance tomography (ECT) system. To achieve both high-quality images and fast imaging speed with limited measurement data, an image reconstruction algorithm, which was initially proposed for compressive sensing, is adapted for ECT image reconstruction to optimize the ill-posed nature of its inverse problem. The proposed algorithm leverages deep learning networks inspired by the iterative shrinkage-thresholding algorithm (ISTA), thereby creating a model that is both mathematically interpretable and endowed with trainable parameters. Building upon this foundation, the conventional Landweber iteration is integrated with the ISTA-Net to refine the optimization process for ECT image reconstruction. In order to propose an effective model adapting to the actual multiphase flow characteristics and complex flow pattern changes, the training and test process is driven by a comprehensive dataset generated from dynamic simulations, rather than artificial samples of multiphase distributions. This numerical methodology simulates the dynamic measurement process of a virtual ECT sensor by coupling the gas–liquid two-phase flow field and the ECT electrostatic field. The results of the testing phase indicate that the proposed algorithm outperforms traditional ECT image reconstruction methods. Compared with the linear back projection algorithm, the average image error and gas fraction error have been reduced by 20.44% and 16.74%, respectively, while maintaining a computational speed comparable to that of the Landweber iteration. The accuracy of the new algorithm in reconstructing the two-phase interface and estimating the gas fraction has been validated by static experimental tests, showing its potential for practical application in online gas–liquid two-phase flow measurement scenarios.

A modified iteratively regularized Landweber iteration method: Hölder stability and convergence rates

Abstract In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable.

Noise propagation in linear stationary iterations

AbstractWe consider a stationary iteration for solving a linear system of arbitrary order. The method includes, e.g. Kaczmarz iteration, the Landweber iteration and the SOR (Gauss-Seidel) iteration. A study of the behavior of the iterates, both theoretically and experimentally, is performed. In particular we compare the behavior with and without noise in the data. The results give insight into the interplay between noise free and noisy iterates. For comparision we also included a Krylov type method CGLS in the experiments. As expected CGLS works well for noise free data but also tends to amplify the noise faster than the other methods, thus making it more critical when to stop the iterations.

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★ ★ The project is supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX23_2549) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190578).

In the article, we focus on the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition on a cylindrical symmetric field. The ill-posedness of this problem is proved. We introduce the modified Landweber iteration method(MLIM), the Truncated singular value decomposition(TSVD) method for solving it and propose a new regularization method, named as the TSVD-modified Landweber iteration method(TMLIM). The error estimates between the exact solution and the regularized approximate solution are presented by using two regularization parameter selection rules. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the regularization methods.

Defect detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method

ABSTRACT Electromagnetic tomography (EMT) uses the mutual inductance of the coil to visualise the conductivity distribution of interesting regions. Since the conductivity of defects and metal samples are different, the metal samples with defects can be treated as binary-valued material distributions. This paper investigates the closed-loop fuzzy proportional, integral and derivative (PID)-controlled iterative Landweber method. The whole method includes fuzzy PID controller, the Landweber reconstruction method, and the Dirichlet-to-Neumann map. Specifically, the differential signal between the mutual inductance of the coil and the feedback signal is used as the input of the fuzzy PID controller. The fuzzy controller can automatically adjust three parameters ( K p , K i and K d ) of PID controller. Subsequently, the output of the PID controller can serve as the input of the Landweber algorithm to reconstruct the distribution of conductivity. Furthermore, the Dirichlet-to-Neumann map is used to calculate the mutual inductance, acting as the feedback signal based on the reconstruction conductivity distribution. Finally, both the numerical simulation and experiments are applied to verify the proposed method. The results indicate that the proposed method can reconstruct the image with a clear edge, and the average correlation coefficient can reach 0.792.

Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary

Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary

带扰动算子的Landweber迭代在Hanke-Raus准则下的收敛阶分析

带扰动算子的Landweber迭代在Hanke-Raus准则下的收敛阶分析

Stochastic linear regularization methods: random discrepancy principle and applications

The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.

Hanke–Raus rule for Landweber iteration in Banach spaces

Hanke–Raus rule for Landweber iteration in Banach spaces

A Preconditioned Landweber iteration-based Bundle adjustment for large-scale 3D reconstruction

Bundle Adjustment (BA) plays a crucial role in 3D reconstruction methods and Structure from Motion (SFM) algorithms, as it involves nonlinear minimization of reprojection error. The efficiency and accuracy of BA solutions are of paramount importance for achieving high-quality 3D reconstructions. The solution of the normal equations derived from nonlinear minimization is typically the most time-consuming step in BA and directly impacts the reconstruction accuracy. In this paper, we propose a Preconditioned Landweber Iteration solver for the normal equations. This solver utilizes a preconditioner and a corresponding relaxation strategy to improve the conditioning of the normal equations, leading to improved convergence speed and accuracy of the iteration. Compared to traditional direct and iterative algorithms, our approach significantly enhances the accuracy and solution time of BA. Experimental results demonstrate that our proposed solver outperforms the Schur complement solver in terms of both speed and accuracy when solving the normal equations. This improvement contributes to further acceleration and accuracy in large-scale 3D reconstruction, particularly for challenging large-scale intensive scenarios.

Solving stationary inverse heat conduction in a thin plate

We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.

The Block Landweber Iterative Method for Light Field Reconstruction from a Focal Stack

Light field imaging involves reconstructing a 4D light field from a 3D focal stack, which makes it challenging to reconstruct the light field from incomplete projection data. To address this problem, a linear projection system is established to model the focal stack imaging process using discrete refocusing equations. Based on this system, we propose the block Landweber iterative method to find the least-squares solution. This method computes the sparse matrix while iterating, which overcomes the problem of data storage. The 2-norm of the block matrix is utilized as the weighted matrix to normalize every block matrix on an identical scale, delivering an effective relaxation strategy under the convergence condition in the inconsistent case, which yields better reconstruction results and accelerates the convergence speed. The experimental results based on the image quality assessments of reference and non-reference images show that our method achieved better reconstruction results compared to other relevant common methods, even with fewer focal stacks and higher angle resolution.

Comparison and application of downward continuation methods for potential field data

Downward continuation highlights the details of the gravity and magnetic anomalies and improves the horizontal resolution of potential field data. However, downward continuation acts as a high-pass filter. As depth increases, any high-frequency noise will be quickly amplified and mask the effective information in the continuation field. We review 13 popular downward continuation methods and categorize them into five classes. The methods are compared and discussed in terms of continued depth and noise robustness using synthetic data and then are applied to field data from the Pobei area in Xinjiang Province, China. It is found that the noise robustness of the integral equation approach is improved using the iterative Landweber form. The frequency-domain methods based on high-frequency truncation do not require iterations and have a fast calculation speed. Using the Laplacian of the Gaussian operator to calculate the second-order vertical derivative enhances the noise robustness of the Taylor series expansion-based techniques. Segmentation of the continued depth followed by downward continuation increases the accuracy of extrapolation methods. High-order derivative terms located in the denominator of the expansion series cause instability in the continuous fractional series expansion-based techniques, and the obtained results have to be further proceeded with median filtering.

Relaxation strategy for the block Landweber method

The block Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the block Landweber iteration is of both theoretical and practical importance. When the linear system is consistent, we derive a neat iterative representation and find an optimal relaxation coefficient by minimizing the 2-norm of the newly derived iteration matrix in the range of the conjugate transpose of the corresponding block matrix. We also establish the accelerating convergence relaxation strategies based on the maximum eigenvalue of each block matrix. When the linear system is inconsistent, we normalize each block matrix by the 2-norms of the each weighted block matrix at a uniform scale to give the relaxation strategies. The numerical simulations show that the relaxation strategies proposed for the block Landweber iteration in both consistent and inconsistent cases in this paper have the better reconstruction performances. Compared with the Landweber iteration, the block Landweber iteration has superiority in reconstruction results.

A 12-Electrode ECT Sensor with Radial Screen for Fluid Diagnosis

In this paper, the use of radial screens in an Electrical Capacitance Tomography (ECT) sensor is investigated with the aim of resulting in higher-resolution images. The effect of the screens on the quality of the reconstructed images was simulated. 12-electrode ECT sensors with and without screens were designed and tested. The capacitance between different electrode pairs was measured for two permittivity distributions. The obtained capacitance data were used to reconstruct images using the Projected Landweber Iteration algorithm based on linear and semi-linear ECT models. The sensitivity of the measurements and the accuracy of the obtained images for the ECT sensors with and without screens were analyzed. The main conclusion is that a little improvement in the quality of images can be achieved with the use of 12-Electrode ECT sensors with radial screen.

Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment

Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

The Landweber Iterative Regularization Method for Identifying the Unknown Source of Caputo-Fabrizio Time Fractional Diffusion Equation on Spherically Symmetric Domain

In this article, the inverse problem for identifying the space-dependent source of time fractional diffusion equation on a spherically symmetric domain with Caputo–Fabrizio fractional derivative is discussed. This problem is a typical ill-posed problem and the Landweber iterative regularization method is used to obtain the approximation solution. The convergent error estimates under a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are given, respectively. The numerical examples are given to show the effectiveness of the Landweber iterative regularization method.