Iterative Regularization Method Research Articles

An Arnoldi-based preconditioner for iterated Tikhonov regularization

Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel “approximate Tikhonov regularization method” based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method.

Iterative hybrid regularization for extremely noisy full models in single particle analysis

Cryo-electron microscopy single particle analysis (SPA) represents a vital tool for structure determination of macromolecules. Discrete inverse problems arising in this field are extremely large and seriously contaminated by noise. The model matrix is highly structured and can not be stored explicitly. Iterative regularization methods are used here only rarely, since it is believed that they are computationally expensive and require complicated stopping criteria. In this paper, we overcome these difficulties and demonstrate that Hybrid Krylov subspace methods can be used to solve SPA inverse problems efficiently. We propose an application-driven regularization parameter selection approach and present a matrix-free implementation of the hybrid solver for GPU computations in single precision arithmetic. In comparison to Fourier-based techniques, the hybrid approach allows to compute reconstructions from full (nonreduced) SPA models even for highly noisy data sets.

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.

A comparison of regularization models for few-view CT image reconstruction

In this paper I analyse some regularization models for the reconstruction of X-rays Computed Tomography images from few-view projections. It is well known that the widely used low-cost Filtered Back Projection method is not suitable in case of low-dose data, since it produces images with noise and artifacts. Iterative reconstruction methods based on the model discretization are preferred in this case. However, since the problem has infinite possible solutions and is ill-posed, regularization is necessary to obtain a good solution. Different iterative regularization methods have been proposed in literature, but an organized comparison among them is not available. We compare some regularization approaches in the case of few-view tomography by means of simulated projections from both a phantom and a real image.

Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model

Abstract This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy’s boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm.

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.

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

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

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.

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

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.

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.

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.

Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.

Fractional Landweber Iterative Regularization Method for Identifying the Unknown Source of the Time-Fractional Diffusion Problem

Fractional Landweber Iterative Regularization Method for Identifying the Unknown Source of the Time-Fractional Diffusion Problem

A fast two-point gradient algorithm based on sequential subspace optimization method for nonlinear ill-posed problems

In this paper, we propose a fast two-point gradient algorithm for solving nonlinear ill-posed problems, which is based on the sequential subspace optimization method. The key idea, in contrast to the standard two-point gradient method, is to use multiple search directions in each iteration without extra computation, and to get the step size by metric projection. Moreover, a modified discrete backtracking search algorithm is proposed to select the combination parameters in the accelerated two-point gradient method. Under the basic assumptions for iterative regularization methods, we establish the convergence results of the method in the noise-free case. Furthermore, stability and regularity are presented when the algorithm terminated by the discrepancy principle for the case of noisy data. Finally, some numerical simulations are presented, which exhibit that the proposed method leads to a significant reduction of the iteration numbers and the overall computational time.

Landweber Iterative Regularization Method for Identifying Unknown Source for the Biharmonic Equation

Landweber Iterative Regularization Method for Identifying Unknown Source for the Biharmonic Equation

A model reduction approach for inverse problems with operator valued data

We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.

Three regularization methods for identifying the initial value of homogeneous anomalous secondary diffusion equation

In this paper, the inverse problem of initial value identification for homogeneous anomalous diffusion equation with Riemann‐Liouville fractional derivative in time is studied. We prove that this kind of problem is ill‐posed. We analyze the optimal error bound of the problem under the source condition and apply the quasi‐boundary regularization method, fractional Landweber iterative regularization method, and Landweber iterative regularization method to solve this inverse problem. Based on the results of conditional stability, the error estimates between the exact solution and the regular solution are given under the priori and posteriori regularization parameter selection rules. Finally, three examples are given to illustrate the effectiveness and feasibility of these methods.

Study on neutron spectrum unfolding method of organic scintillation measurement based on iterative regularization

An iterative regularization method is proposed for neutron spectrum unfolding of the liquid organic scintillator EJ-301. The Arnoldi iteration is used to generate Krylov subspaces. GMRES (Generalized Minimal Residual) combined with Tikhonov regularization is used for neutron spectrum updating. The restart method is applied to speed up the convergence as well as non-negativity constraints. The Monte Carlo code Geant4 is applied to simulate the response functions and PHSs (pulse height spectrums) of the neutron spectrums for unfolding. Four typical neutron spectrums are studied, such as the mono-energetic DD fusion neutron spectrum, 241Am-Be neutron spectrum, 252Cf neutron spectrum and Gaussian broadened DT fusion neutron spectrum. Comparison study between the proposed method and the existing methods is also carried out, such as GRAVEL, MLEM, and direct Tikhonov regularization. Unfolding results with acceptable accuracies are obtained with the proposed method, and the proposed method shows better adaptabilities than the other three methods with both the ideal and simulated PHSs in most cases.