Iterative Regularization Research Articles

The Landweber Iterative Regularization Method for Solving the Cauchy Problem of the Modified Helmholtz Equation

In this manuscript, the Cauchy problem of the modified Helmholtz equation is researched. This inverse problem is a serious ill-posed problem. The classical Landweber iterative regularization method is designed to find the regularized solution of this inverse problem. The error estimations between the exact solution and the regularization solution are all obtained under the a priori and the a posteriori regularization parameter selection rule. The Landweber iterative regularization method can also be applied to solve the Cauchy problem of the modified Helmholtz equation on the spherically symmetric and cylindrically symmetric regions.

On the Iterative Regularization of the Lagrange Principle in Convex Optimal Control Problems for Distributed Systems of the Volterra Type with Operator Constraints

On the Iterative Regularization of the Lagrange Principle in Convex Optimal Control Problems for Distributed Systems of the Volterra Type with Operator Constraints

A SAR change detection method based on an iterative guided filter and the log mean ratio

ABSTRACT Change detection is an essential step for synthetic aperture radar (SAR) image interpretation. We propose a change detection method that included a denoising method and a difference operator to improve the change precision. The SAR images are first transformed into logarithmic domain before filtering. And then, the log-intensity image is filtered by the iterative guided filter that is the combination of a traditional guided filter and iterative regularization technique. The operator based on the log-ratio (LR) and neighbourhood pixels is developed to generate the difference image, which can reflect the change trends between images effectively. The final image that consisted of changed pixels and unchanged pixels is obtained by using the fuzzy c-means (FCM) algorithm. The experimental results accomplished over two real SAR datasets demonstrate that the proposed method can improve the accuracy of change detection.

Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method

Fluid-saturated porous media are two-phase media, which are composed of solid and liquid phases. Biot theory for fluid-saturated porous media holds that underground media are composed of porous elastic solid and compressible viscous fluid filled with pore space. Compared with the single-phase media theory, the fluid-saturated porous media theory can describe the subsurface media more precisely, and the elastic wave equations in the fluid-saturated porous media contain more parameters used to describe the formation properties. Therefore, fluid-saturated porous media theory is widely used in geophysical exploration, seismic engineering, and other fields.This paper considers the porosity reconstruction problem of the elastic wave equations in the fluid-saturated porous media based on Biot theory, which is a typically nonlinear ill-posed inverse problem from mathematical viewpoint. The proposed method is a novel iteration regularization scheme based on the homotopy perturbation technique. To verify the validity and applicability, numerical experiments of two-dimensional and three-dimensional porosity models have been carried out. Numerical results illustrate that this method can overcome the numerical instability and are robust to data noise in the reconstruction procedure. Furthermore, compared with the classical regularized Gauss-Newton method, the homotopy perturbation method greatly widens the convergence region while keeping the fast convergence rate.

Iteration regularization method for a sideways problem of time-fractional diffusion equation

Iteration regularization method for a sideways problem of time-fractional diffusion equation

Regularizing ill-posed problem of single-epoch precise GNSS positioning using an iterative procedure

Abstract This paper analyses the regularization of an ill-conditioned mathematical model in a single-epoch precise GNSS positioning. The regularization parameter (RP) is selected as a parameter that minimizes the criterion of the Mean Squared Error (MSE) function. The crucial for RP estimation is to ensure stable initial least-squares (LS) estimates to replace the unknown quadratic matrix of actual values with the LS covariance matrix. For this purpose, two different data models are proposed, and two research scenarios are formed. Two regularized LS estimations are tested against the non-regularized LS approach. The first one is the classic regularization of LS estimation. In turn, the second one is its iterative counterpart. For the LS estimator of iterative regularization, regularized bias is significantly lower while the overall accuracy is improved in the sense of MSE. The regularized variance-covariance matrix of better precision can mitigate the impact of regularized bias on integer least-squares (ILS) estimation up to some extent. Therefore, iterative LS regularization is well-designed for single-epoch integer ambiguity resolution (AR). Nevertheless, the performance of the ILS estimator is studied in the context of the probability of correct integer AR in the presence of regularized bias.

Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation

The main purpose of this paper is to identify simultaneously a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation from two points observed data. First of all, using the fixed point theorem, we prove the existence and uniqueness of the solution for the direct problem. Secondly, the stability of the inverse problem is proved and the uniqueness is a direct result of the stability estimate. In addition, we illustrate the ill-posedness of the inverse problem and use a non-stationary iterative Tikhonov regularization method to recover numerically the time dependent potential coefficient and source term. At the same time, we give the existence of the minimizer for the minimization functional. In order to solve the minimization problem, we apply an alternating minimization method to find the minimizer and prove the solving sub-problems are stable on noisy data as well as prove the data fidelity item decreases monotonously with the iterative running. Finally, some numerical examples are provided to shed light on the validity and robustness of the numerical algorithm.

Online Optimization with Feedback Delay and Nonlinear Switching Cost

We study a variant of online optimization in which the learner receives k-rounddelayed feedback about hitting cost and there is a multi-step nonlinear switching cost, i.e., costs depend on multiple previous actions in a nonlinear manner. Our main result shows that a novel Iterative Regularized Online Balanced Descent (iROBD) algorithm has a constant, dimension-free competitive ratio that is $O(L^2k )$, where L is the Lipschitz constant of the switching cost. Additionally, we provide lower bounds that illustrate the Lipschitz condition is required and the dependencies on k and L are tight. Finally, via reductions, we show that this setting is closely related to online control problems with delay, nonlinear dynamics, and adversarial disturbances, where iROBD directly offers constant-competitive online policies.

High-Quality Self-contained Electromagnetic Imaging Scheme Based on Projected Nonlinear Landweber and Machine Learning

This work aims to build an inverse electromagnetic imaging scheme exploiting the best of the following two methodologies: iterative nonlinear regularization and recently developed machine learning models. The proposed method is designed to achieve three targets: 1) the elimination of user intervention to empirically select inverse parameters; 2) high-quality image recovery; and 3) stable scheme with accelerated reconstruction time. To achieve these targets, an iterative imaging algorithm is developed based on projected nonlinear Landweber (PNLW) supported by three neural networks that: 1) predict first-norm solutions, which is an inverse parameter required by the projection operator; 2) predict iteration step lengths that ensure PNLW convergence; and 3) filter recovered images via PNLW to enhance image quality. The first-norm projection operator satisfies the sparse constraint. The proposed scheme is efficient for applications where the number of measurements is limited by adopting sparse optimization. The results of the proposed scheme and conventional approach are compared in numerical tests to assess the efficiency and performance of the proposed scheme.

Phase Design and Near-Field Target Localization for RIS-Assisted Regional Localization System

This paper explores the near-field regional target localization problem with the reconfigurable intelligent surface (RIS) assisted system. Traditional near-field localization strategies typically estimate the position of the target through line-of-sight (LOS) signals transmitted from the anchor node. In practice, accurate location estimation is a challenging issue when the LOS link between the anchor node and target may be unavailable due to the obstacle. To this end, this paper investigates the possibility to confirm the position of the target node consisted in an area of interest (AOI) by retrieving information from the RIS reflection signals. Specifically, we establish a general framework for RIS-assisted regional localization, which consists of RIS phase design and position determination. By defining the average localization accuracy (ALA) of the AOI, we first present a discretization method to design the RIS phase. Then, a robust phase design problem is formulated via transforming the AOI into an uncertainty model of position parameters, and an efficient iterative entropy regularization (IER)-based algorithm is proposed to solve it. Using the designed RIS phase, we develop a near-field target localization algorithm and discuss the power optimization problem for the RIS-assisted localization system (RALS). Numerical results demonstrate the effectiveness of the proposed framework, in which both phase design strategies almost coincide with the optimal RIS phase, and the proposed localization method can attain the near-optimal localization performance by applying the designed RIS phase schemes.

Разработка комплекса параллельных программ решения обратных задач гравиметрии и магнитометрии для сеток большой размерности

The most important problems in studying of the structure of the earth’s crust are the inverse problems of gravimetry and the problems of magnetometry. The problem is in finding interfaces between layers with different constant densities using known gravitational data. The methods for solving these problems are based on the ideas of iterative regularization. After discretization, these are reduced to systems of nonlinear functions of large dimensions. The need to improve the accuracy of the results of solving problems entails an increase in the computation time. The paper describes a remote computing system and an integrated program complex for graphics accelerators that implement the fastest and most memory-efficient algorithms developed earlier and based on gradient methods. The remote computing system considered in the work is a web portal that is a universal solution for launching tasks on remote clusters. The most important advantage of the portal is the simplicity of its use: when connecting to a cluster to carry out calculations, you no longer need to install additional software on the cluster itself, and you do not need a privileged account to work with the cluster. All that is required is a valid account on the remote cluster, the rest of the work on communication with the data processing center (DPC) is taken over by the portal. The portal can easily scale with the growth of the number of users who can download the necessary algorithms and perform calculations using the data center.

A Study of a Posteriori Stopping in Iteratively Regularized Gauss–Newton-Type Methods for Approximating Quasi-Solutions of Irregular Operator Equations

A Study of a Posteriori Stopping in Iteratively Regularized Gauss–Newton-Type Methods for Approximating Quasi-Solutions of Irregular Operator Equations

Damage identification of structural systems by modal strain energy and an optimization-based iterative regularization method

Damage identification of structural systems by modal strain energy and an optimization-based iterative regularization method

Extended Distribution of Relaxation Time Analysis for Electrochemical Impedance Spectroscopy

The distribution of relaxation time (DRT) is increasingly investigated as a novel analytical method for electrochemical impedance spectroscopy. However, this method has not yet been generalized, as it cannot be applied to a spectrum influenced by an isolated capacitator and a resistor connected in parallel with an inductor. To generalize the DRT analysis, we propose a novel principal relation and actual calculation methods using an iterative elastic net regularization algorithm. The elastic net regularization was implemented using the Levenberg–Marquardt method. The proposed algorithm aids in analyzing the DRT for spectra that cannot be solved using conventional methods. We compared the DRT results in different regularization methods obtained from the proposed program with those obtained from other open-source programs. Additionally, the realistic problems of the DRT analysis are discussed considering the calculated results and their theoretical basis. Different DRT curves can be obtained depending on the particular program and regularization methods, even when the spectrum is the same. Moreover, overconfidence in the electrochemical DRT method can be avoided with a clear understanding of DRT basics.

Efficient Iterative Regularization Method for Total Variation-Based Image Restoration

Total variation (TV) regularization has received much attention in image restoration applications because of its advantages in denoising and preserving details. A common approach to address TV-based image restoration is to design a specific algorithm for solving typical cost function, which consists of conventional ℓ2 fidelity term and TV regularization. In this work, a novel objective function and an efficient algorithm are proposed. Firstly, a pseudoinverse transform-based fidelity term is imposed on TV regularization, and a closely-related optimization problem is established. Then, the split Bregman framework is used to decouple the complex inverse problem into subproblems to reduce computational complexity. Finally, numerical experiments show that the proposed method can obtain satisfactory restoration results with fewer iterations. Combined with the restoration effect and efficiency, this method is superior to the competitive algorithm. Significantly, the proposed method has the advantage of a simple solving structure, which can be easily extended to other image processing applications.

Synthetic Aperture Radar Image Despeckling Based on Multi-Weighted Sparse Coding.

Synthetic aperture radar (SAR) images are inherently degraded by speckle noise caused by coherent imaging, which may affect the performance of the subsequent image analysis task. To resolve this problem, this article proposes an integrated SAR image despeckling model based on dictionary learning and multi-weighted sparse coding. First, the dictionary is trained by groups composed of similar image patches, which have the same structural features. An effective orthogonal dictionary with high sparse representation ability is realized by introducing a properly tight frame. Furthermore, the data-fidelity term and regularization terms are constrained by weighting factors. The weighted sparse representation model not only fully utilizes the interblock relevance but also reflects the importance of various structural groups in despeckling processing. The proposed model is implemented with fast and effective solving steps that simultaneously perform orthogonal dictionary learning, weight parameter updating, sparse coding, and image reconstruction. The solving steps are designed using the alternative minimization method. Finally, the speckles are further suppressed by iterative regularization methods. In a comparison study with existing methods, our method demonstrated state-of-the-art performance in suppressing speckle noise and protecting the image texture details.

An Iterative Regularization Method Based on Tensor Subspace Representation for Hyperspectral Image Super-Resolution

Hyperspectral image super-resolution (HSI-SR) can be achieved by fusing a paired multispectral image (MSI) and hyperspectral image (HSI), which is a prevalent strategy. But, how to precisely reconstruct the high spatial resolution hyperspectral image (HR-HSI) by fusion technology is a challenging issue. In this paper, we propose an iterative regularization method based on tensor subspace representation (IR-TenSR) for MSI-HSI fusion, thus HSI-SR. First, we propose a tensor subspace representation (TenSR)-based regularization model that integrates the global spectral-spatial low-rank and the nonlocal self-similarity priors of HR-HSI. These two priors have been proven effective, but previous HSI-SR works cannot simultaneously exploit them. Subsequently, we design an iterative regularization procedure to utilize the residual information of acquired low-resolution images, which are ignored in other works that produce suboptimal results. Finally, we develop an effective algorithm based on the proximal alternating minimization method to solve the TenSR-regularization model. With that, we obtain the iterative regularization algorithm. Experiments implemented on the simulated and real datasets illustrate the advantages of the proposed IR-TenSR compared with state-of-the-art fusion approaches. The code is available at https://github.com/liangjiandeng/IR-TenSR.

Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions

The source term identification of the time-fractional diffusion equation with Robin boundary conditions is studied. This problem is ill-posed. Therefore, an iterative regularization method is constructed to calculate the regularized approximate solution of the source term. The error estimates between the regularized approximate solution and the exact solution are given under the priori and the posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the method.

The domain derivative for semilinear elliptic inverse obstacle problems

<p style='text-indent:20px;'>We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.</p>

A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.