Landweber Iteration Research Articles

Spectral algorithms for learning with dependent observations

We study a class of spectral learning methods with dependent observations, including popular ridge regression, Landweber iteration, spectral cut-off and so on. We derive an explicit risk bound in terms of the correlation of the observations, regularity of the regression function, and effective dimension of the reproducing kernel Hilbert space. By appropriately choosing regularization parameter according to the sample size, the risk bound yields a nearly optimal learning rate with a logarithmic term for strongly mixing sequences. We thus extend the applicable range of spectral algorithm to non-i.i.d. sampling process. Particularly, it is shown that the learning rates for i.i.d. samples in the literature refer to our special case, i.e., the mixing condition parameter tends to zero.

Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces

Abstract In this paper, we study the convergence analysis of the inexact Newton–Landweber iteration method (INLIM) with frozen derivative in Hilbert as well as Banach spaces. To study the convergence analysis, we incorporate the Hölder stability of the inverse mapping and Lipschitz continuity of the Fréchet derivative of the forward mapping. Moreover, we derive the convergence rates of INLIM in Hilbert as well as Banach spaces without using any extra smoothness condition. Finally, we compare our convergence rates results with that of several other frozen methods proposed in the literature to solve inverse problems.

Robot sensing based on electrical capacitance tomography sensor with rotation

The existing sensing technology for industrial robot hands cannot sense inside of an object. To recognise the material and internal distribution of an object when it is grasped, electrical capacitance tomography (ECT) may be used if the object is non-conductive, as ECT can measure the permittivity distribution inside an object from external capacitance measurements. An ECT sensor normally has 8 or 12 electrodes equally distributed around an object. This paper introduces the first attempt to make use of ECT with robot hands. Because the number of electrodes is limited, it is necessary to rotate the object to be imaged mechanically, so that sufficient measurements can be taken. Four aspects of the ECT sensor are discussed based on simulation: (1) the structure of electrodes; (2) the number of rotation times; (3) the starting position of rotation; and (4) the diameter of rotation. Two image reconstruction algorithms are used: the linear-back projection and Landweber iteration with three typical distributions. The results show that the rotational three-electrode ECT structure has good ability to produce useful images, showing the internal distribution of unknown objects.

Heuristic rule for inexact Newton-Landweber iteration with convex penalty terms of nonlinear: ill-posed problems

Inexact Newton regularization methods are a family of prominent regularization methods for solving nonlinear ill-posed problems, which consist of an outer Newton iteration and an inner scheme providing increments by applying the regularization technique to the local linearized equations. In this paper, we propose a heuristic stopping rule for the inexact Newton regularization method, where the inner scheme is defined by Landweber iteration and the strong convex function is incorporated as the penalty term. In contrast to a prior and a posteriori stopping rules, our heuristic rule is purely data driven and does not require the information on noise level, which renders the method feasible when the noise level is unknown or unreliable. Under certain assumptions on the random noise, we establish a new convergence analysis for the inexact Newton-Landweber method under the heuristic rule. The numerical simulations are provided to demonstrate the performance of our heuristic rule.

Solution of the backward problem for the space-time fractional diffusion equation related to the release history of a groundwater contaminant

Abstract Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms.

A Study of an Iteratively-Regularized Simplified Landweber Iteration for Nonlinear Inverse Problems in Hilbert Spaces

Lanweber-type methods are a well-known regularization methods to solve linear and nonlinear ill-posed problems. In this article, we consider a simplified form of Landweber method, say, an iteratively-regularized simplified Landweber iteration for solving nonlinear ill-posed problems. We study a detailed convergence analysis of the method under standard conditions on the nonlinearity and the rate of convergence under a Hölder-type source condition. Finally, numerical simulations are performed to validate the performance of the method.

The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

Image reconstruction based on a modified bird swarm optimization algorithm for electrical impedance tomography

Electrical impedance tomography (EIT) is a potential and promising tomographic technique. It is known that image reconstruction of EIT is mathematically an ill-posed inverse problem that leads to poor reconstruction quality. To improve the quality of reconstructed images, a novel image reconstruction method based on modified bird swarm optimization algorithm is proposed in this paper. An image of conductivity distribution acquired by Newton's one-step error reconstruction algorithm performs as initial population. Then a modified bird swarm optimization algorithm is proposed to optimize the initial conductivity distribution. By introducing a dynamic inertia weight parameter, flight interval of the bird swarm varies with the number of iterations. With the proposed method, a strong global search capability in the initial stage and a strong local search capability in the later stage are ensured during the optimization. To verify the effectiveness of the proposed method, simulation work and experimental validation are carried out. Also, reconstructed images are compared with the results obtained by iterative Landweber method and Newton–Raphson method. Meanwhile, the impact of noise on the reconstruction is investigated. The simulation and experimental results have demonstrated the superior performance of the proposed method in visualizing conductivity distribution.

Investigation of the well-posedness of the cauchy problem for the helmholtz equation

In this paper we consider the the initial boundary value problem for the Helmholtz equation. The well-possedness of the Cauchy problem for the Helmholtz equation is investigated and the uniqueness of the solution of the original problem in the class of functions represented as Fourier series is proved. For a direct problem, a proof of the stability theorem of the generalized solution is presented, and an estimate of the stability of the generalized solution is obtained. To solve the problem, the original problem is reduced to the inverse problem and is written in operator form. We reduce this problem to the functional minimization problem, the Landweber iteration method is used in the minimization problem. A numerical study of the stability of the direct and initial problems is carried out. The results of the study are shown in tables and figures.

Improved local convergence analysis of the Landweber iteration in Banach spaces

Improved local convergence analysis of the Landweber iteration in Banach spaces

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton–Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

A Trained Iterative Shrinkage Approach Based on Born Iterative Method for Electromagnetic Imaging

A trained-based Born iterative method (TBIM) is developed for electromagnetic imaging (EMI) applications. The proposed TBIM consists of a nested loop; the outer loop executes TBIM iteration steps, while the inner loop executes a trained iterative shrinkage thresholding algorithm (TISTA). The applied TISTA runs linear Landweber iterations implemented with a trained regularization network designed based on U-net architecture. A normalization process was imposed in TISTA that made TISTA training applicable within the proposed TBIM. The iterative utilization of the regularization network in TISTA is a bottleneck that demands high memory allocation through the training process. Therefore TISTA within each TBIM step was trained separately. The TISTA regularization network in each TBIM step was initialized using the weights from the previous TBIM step. The above approach achieved high-quality image restoration after running few TBIM steps while maintained low memory allocation through the training process. The proposed framework can be extended to Newton or quasi-Newton schemes, where within each Newton iteration, a linear ill-posed problem is optimized that differs from one example to another. The numerical results illustrated in this work show the superiority of the proposed TBIM compared to the conventional sparse-based Born iterative method (SBIM).

Fractional Landweber Iterative Regularization Method for Solving the Inverse Problem of Time-Fractional Schrödinger Equation

In this paper, we consider the inverse problem for the time-fractional Schrödinger equation. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. We use the fractional Landweber iterative regularization method to solve this inverse problem and obtain the regularization solution. Under a priori and a posteriori regularization parameter choices, the error estimates are all obtained, respectively.

Experimental imaging and algorithm optimization based on deep neural network for electrical capacitance tomography for LN2-VN2 flow

The deep neural network (DNN) method is applied to better map the phase distribution compared with the conventical reconstruction algorithm for nonlinear problems of electrical capacitance tomography (ECT). Three DNNs are trained and evaluated in the numerical experiment for cryogenic two-phase flow. A capacitance acquisition unit is designed. And a cryogenic experiment with liquid nitrogen and nitrogen vapor (LN2-VN2) as the working fluids is conducted to verify the imaging solution for practical application. The DNN with the input of the measured capacitance data (DNN-C) has shown its generalization ability for the practical imaging application. Further, the DNN optimal method is coupled to the conventional LBP and Landweber iteration algorithm by evaluating the nonlinearity between the error capacitance vector and the real phase distribution vector (DNN-EC). The comparison shows that the DNN modification method can significantly improve the phase distribution closed to the pattern in the training input samples with an elimination of the artifacts and sharpening interface, but it fails to recognize the flow pattern which is not included in the samples. The result shows that compared with the DNN-EC, DNN-C is a better choice for the image reconstruction for the cryogenic application.

Updating the Landweber Iteration Method for Solving Inverse Problems

The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP).

Black-box image deblurring and defiltering

Given an image filter, defiltering refers to the problem of recovering an original image from its filtered version, assuming that the internal structure of the filter is not known. In this paper, we propose five iterative image defiltering schemes and use them for a semi-blind image deblurring problem. Namely, given an image resulting from applying a blurring image filter corrupted by noise to a clean image, we use the proposed iterative schemes to achieve a restoration of the clean image. In particular, for the motion deblurring problem, we show that our defiltering schemes are competitive with modern non-blind image deconvolution methods while using less information. The schemes are inspired by classical methods solving inverse problems and consist of properly modified and extended versions of the Van Cittert iterations, Levenberg–Marquardt method, Wiener filter, Landweber iterations, and Richardson–Lucy algorithm. In addition to dealing with image deblurring problems, we show that the proposed schemes can be used for inverting non-linear filters, and show that they are competitive with state-of-the-art black-box defiltering methods for these problems.

Coupling RBF-based meshless method and Landweber iteration algorithm for approximating a space-dependent source term in a time fractional diffusion equation

The problem of determining a space-dependent source term in a time fractional diffusion equation is considered from the measured data at the final time. To find an approximation of the source term, a methodology involving minimization of the cost functional is applied. Also, in order to construct the Landweber iteration algorithm, an explicit formula for the gradient of the cost functional J is given via the solution of an adjoint problem. The resulting adjoint problem is treated by a radial basis function method for spatial dimension and a finite difference scheme for the time fractional derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of Landweber iteration algorithm are proved. At the end, a numerical example is given to show the validation of the iterative algorithm.

Reverse image filtering with clean and noisy filters

Given an image filter {{varvec{y}}}={{varvec{f}}},({{varvec{x}}}), where {{varvec{x}}} and {{varvec{y}}} are input and output images, respectively, reverse image filtering consists of rendering an approximation to {{varvec{x}}} from {{varvec{y}}} using the filter {{varvec{f}}},(cdot ) itself as a black box, without knowing the internal structure of the filter. In this paper, we propose to use modified Landweber iterations for reverse image filtering, evaluate the performance of our approach, and present applications to image deblurring and super-resolution. An important advantage of our approach over the existing reverse image filtering methods is high robustness to noise.

Landweber Iteration-Inspired Network for Image Super-Resolution

Image super-resolution (SR) is one of the classical ill-posed image processing issues to generate high-resolution (HR) images from given low-resolution (LR) instances. Recent SR works aim to find an elaborate convolutional neural network (CNN) design and regard it as an end-to-end filter to map the image from LR space to HR space. However, seldom of them concentrate on the mathematical proof of network design or consider the problem from an optimization perspective. In this paper, we investigate the image SR based on the Landweber iteration method, which is an effective optimization method to find a feasible solution for the ill-posed problem. By considering the issue from the optimization perspective, we design a corresponding Landweber iteration-inspired network to adaptively learn the parameters and find the HR results. Experimental results show the proposed network achieves competitive or better subjective and objective performance than other state-of-the-art methods with fewer parameters and computational costs.

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.