Iterative Regularization Method Research Articles

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.

Refined Solution of the Variational Problem of Identification of Lumped Parameter Mathematical Models of Heat Transfer

A refined solution of the variational problem of determining the step of descent of the iterative regularization method is found within the problem of mathematical reliability of the parametric identification of lumped parameter mathematical heat transfer models. The results are used to identify the mathematical thermal model of part of spacecraft and are confirmed by calculation experiments.

Expanding the applicability of an iterative regularization method for ill-posed problems

Expanding the applicability of an iterative regularization method for ill-posed problems

A new hybrid conjugate gradient method for dynamic force reconstruction

A new hybrid conjugate gradient method is proposed in this article based on the gradient operator and applied to the structural dynamic load identification problem. It has proved that the present method with the strong Wolfe line search possesses sufficient descent property. In addition, the present method is globally convergent when the parameter in the strong Wolfe line search conditions is restricted in some suitable intervals. Three example problems from engineering are solved by the newly developed conjugate gradient method to demonstrate the robustness and effectiveness of conjugate gradient method in solving multi-source dynamic load identification problems. Compared with the traditional Landweber iteration regularization method (Landweber), the proposed conjugate gradient method can more stably and effectively overcome the influences of noise, largely reduce the number of iterations, and provide accurate results in identifying multi-source dynamic force in practical engineering structure.

An iterative approach to monochromatic phaseless inverse scattering

This paper is concerned with the inverse problem to recover a compactly supported Schrödinger potential given the differential scattering cross section, i.e. the modulus, but not the phase of the scattering amplitude. To compensate for the missing phase information we assume additional measurements of the differential cross section in the presence of known background objects. We propose an iterative scheme for the numerical solution of this problem and prove that it converges globally of arbitrarily high order depending on the smoothness of the unknown potential as the energy tends to infinity. At fixed energy, however, the proposed iteration does not converge to the true solution even for exact data. Nevertheless, numerical experiments show that it yields remarkably accurate approximations with small computational effort even for moderate energies. At small noise levels it may be worth to improve these approximations by a few steps of a locally convergent iterative regularization method, and we demonstrate to which extent this reduces the reconstruction error.

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem’s coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.

Denoising Weighting Beamforming Method Applied to Sound Source Localization With Airflow Using Microphone Array

Conventional frequency domain beamforming (FDBF) relies on the measured cross-spectral matrix (CSM). However, in wind tunnel tests, the CSM diagonal is contaminated by the interference of incoherent noise after long-time averaging which leads the source map to poor resolution. Diagonal removal (DR) can suppress the noise in beamforming results via the deletion of CSM diagonal, but this method leads to the underestimation of source levels and some negative powers in source maps. Some advanced methods, such as background subtraction, make use of background noise reference to counteract the effects of contamination; however, the results usually become unreliable, because the background noise is difficult to keep constant in different measurements. Diagonal denoising (DD) beamforming is a recent approach to suppress the contamination effects, but it attenuates the noise suppression performance. To overcome the limitations of the above methods, a new method called denoising weighting beamforming (DWB) is proposed in this study on the basis of CSM DD and an iterative regularization method is applied to solve the acoustical inverse problem. Besides, in order to correct the phase mismatch caused by the influence of flow on sound propagation, the shear flow correction is added before using DWB. Experiments on sound source reconstruction are conducted in the environment with the flow. Acoustics data obtained via this method show the successful removal of incoherent noise and the corrected phase mismatch. Furthermore, the sound source localization results are promising and the proposed method is simple to implement.

On estimation of stopping criteria for iterative solutions of gravity downward continuation

This article discusses methods for selecting criteria used for stopping iterative regularization methods applied to solving continuation of gravity observed at or outside the Earth’s surface down to the geoid. This computational step is required for determination of the gravimetric geoid by the Stokes approach. For surface gravity data measured with spatial resolutions at the kilometer level and higher, their downward continuation becomes ill posed and its solution numerically unstable. Two iterative methods often used for solving this problem, namely the Landweber and conjugate gradient least-squares methods, are investigated using a sample of surface gravity data synthesized from a global gravitational model over a mountainous test area in Colorado, USA. Five different commonly used stopping criteria are applied to both iterative methods and neither of them, except L-Curve, succeed to stop the iterative process at the reasonable solution. A simple approach is proposed to stop the iterative methods explo...

A new fast convergent iteration regularization method

Dynamic loads widely exist in numerous applications of mechanical engineering structures, and their identification signifies an important issue for studying mechanical consequence of engineering structures and characterizing their dynamic characteristics. This research aims to reconstruct the dynamic loads in the deterministic structure of thin-walled cylindrical shell and airfoil structure. A new fast convergent iteration regularization method is developed and proposed for identifying dynamic loads. The stability, effectiveness, and convergence of this method are proved according to the regularization theory. The optimum asymptotic convergence order of the regularized solution is also provided according to the Morozov’s discrepancy principle. In addition, two engineering examples are investigated to validate the effectiveness of the identification method proposed. The results demonstrate the applicability of the proposed method in mechanical engineering.

The Averaged Kaczmarz Iteration for Solving Inverse Problems

We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or non-linear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz method. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint sub-problem per iterative update. On the other, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

Optimal Adaptation for Early Stopping in Statistical Inverse Problems

For linear inverse problems Y = Aµ + ξ, it is classical to recover the unknown signal µ by iterative regularization methods (µ (m) , m = 0, 1,. . .) and halt at a data-dependent iteration τ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squarederror A(µ (τ) − µ) 2 is controlled. In the context of statistical estimation with stochastic noise ξ, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squarederror E µ (τ) − µ 2. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L 2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.

Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

We consider a problem of quantitative static elastography, the estimation of the Lam\'e parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lam\'e parameters from displacement data simulating a static elastography experiment are presented.

An iterative method for an inverse source problem of time-fractional diffusion equation

We propose an iteration method to recover a space-dependent source for a time-fractional diffusion equation from the final measurement. Based on the conditional stability of the inverse problem, we prove the convergence of the iterative regularization method under the a priori parameter choice rule and the a posteriori parameter choice rule, respectively. Numerical examples in one dimension and two dimension are given to validate the effectiveness of the presented method.

Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion equation

In this paper, we investigate an inverse problem to determine an unknown source term that has a separable-variable form in the time-fractional diffusion equation, whereby the data is obtained at a certain time. This problem is ill-posed, and we use the Landweber iterative regularization method to solve this inverse source problem. Two kinds of convergence rates are obtained by using an a priori and an a posteriori regularization parameters choice rules, respectively. Numerical examples are provided to show the effectiveness of the proposed method.

A weighted denoising method based on Bregman iterative regularization and gradient projection algorithms

A weighted Bregman-Gradient Projection denoising method, based on the Bregman iterative regularization (BIR) method and Chambolle’s Gradient Projection method (or dual denoising method) is established. Some applications to image denoising on a 1-dimensional curve, 2-dimensional gray image and 3-dimensional color image are presented.Compared with the main results of the literatures, the present numerical results of the proposed method are improved.

Landweber iterative method for identifying a space-dependent source for the time-fractional diffusion equation

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation with variable coefficients in a general bounded domain. This is an ill-posed problem. Firstly, we obtain a regularization solution by the Landweber iterative regularization method. The convergence estimates between regularization solution and exact solution are given under a priori and a posteriori regularization parameter choice rules, respectively. The convergence estimates we obtain are optimal order for any p in two parameter choice rules, i.e., it does not appear to be a saturating phenomenon. Finally, the numerical examples in the one-dimensional and two-dimensional cases show our method is feasible and effective.

Iterative methods for photoacoustic tomography in attenuating acoustic media

The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analysis and a conditional stability result are given. This problem is ill-posed and Landweber iteration regularization method is used to deal with this problem. Convergence estimates are presented under the priori and the posteriori regularization choice rules. For the a priori and the a posteriori regularization parameters choice rules, the convergence error estimates are all order optimal. Numerical examples are given to show that the regularization method is effective and stable for dealing with this ill-posed problem.

Inexact Iterative Bregman Method for Optimal Control Problems

ABSTRACTIn this article, we investigate an inexact iterative regularization method based on generalized Bregman distances of an optimal control problem with control constraints. We show robustness and convergence of the inexact Bregman method under a regularity assumption, which is a combination of a source condition and a regularity assumption on the active sets. We also take the discretization error into account. Numerical results are presented to demonstrate the algorithm.

Self-prior strategy for organ reconstruction in fluorescence molecular tomography.

The purpose of this study is to propose a strategy for organ reconstruction in fluorescence molecular tomography (FMT) without prior information from other imaging modalities, and to overcome the high cost and ionizing radiation caused by the traditional structural prior strategy. The proposed strategy is designed as an iterative architecture to solve the inverse problem of FMT. In each iteration, a short time Fourier transform (STFT) based algorithm is used to extract the self-prior information in the space-frequency energy spectrum with the assumption that the regions with higher fluorescence concentration have larger energy intensity, then the cost function of the inverse problem is modified by the self-prior information, and lastly an iterative Laplacian regularization algorithm is conducted to solve the updated inverse problem and obtains the reconstruction results. Simulations and in vivo experiments on liver reconstruction are carried out to test the performance of the self-prior strategy on organ reconstruction. The organ reconstruction results obtained by the proposed self-prior strategy are closer to the ground truth than those obtained by the iterative Tikhonov regularization (ITKR) method (traditional non-prior strategy). Significant improvements are shown in the evaluation indexes of relative locational error (RLE), relative error (RE) and contrast-to-noise ratio (CNR). The self-prior strategy improves the organ reconstruction results compared with the non-prior strategy and also overcomes the shortcomings of the traditional structural prior strategy. Various applications such as metabolic imaging and pharmacokinetic study can be aided by this strategy.