Iterative Tikhonov Regularization Method Research Articles

A non-stationary iterative Tikhonov regularization method for simultaneous inversion in a time-fractional diffusion equation

The present paper is devoted to recovering the source term and the initial data simultaneously for a time-fractional diffusion equation from additional temperature data at two fixed times t=T1 and t=T2. Firstly, we prove the uniqueness result for the direct problem. And then we apply a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm. Three numerical examples in both one-dimensional and two-dimensional cases are provided to demonstrate the effectiveness and feasibility of the proposed method.

Posterior regularization method for phase removal of shale nano-structure imaging in space domain

X-Ray computed tomography is a non-destructive method that is used, among many applications, to study the size, shape, 3D structures and interconnections of pores in shale. We use phase retrieval methods to deal with the “edge enhancement” effect caused by phase shift. The process of phase retrieval can be described by the transport-of-intensity equation (TIE). But this is an ill-posed problem. The existing methods focus on phase retrieval in the frequency domain. To tackle the ill-posedness, we propose a new method whose main idea is to solve this problem in space domain with a regularization technique. We study a synthetic shale model and simulate the projection data. Then we apply three methods to retrieve the phase: conventional method in frequency domain, direct solving method and iterative Tikhonov regularization method in space domain. Finally, we use the standard filtered back-projection (FBP) method to present the outcome. By analyzing the results, we find advantages of the new method: more stability and fewer artifacts under noise perturbations. The study shows that relative errors of the new method are nearly 1% of that of the traditional method based on frequency domain, and hence the new method is promising for the practical data processing.

Inverse potential problem for a semilinear generalized fractional diffusion equation with spatio-temporal dependent coefficients

In this work, we are interested in an inverse potential problem for a semilinear generalized fractional diffusion equation with a time-dependent principal part. The missing time-dependent potential is reconstructed from an additional integral measured data over the domain. Due to the nonlinearity of the equation and arising of a space-time dependent principal part operator in the model, such a nonlinear inverse problem is novel and significant. The well-posedness of the forward problem is firstly investigated by using the well known Rothe’s method. Then the existence and uniqueness of the inverse problem are obtained by employing the Arzelà–Ascoli theorem, a coerciveness of the fractional derivative and Gronwall’s inequality, as well as the regularities of the direct problem. Also, the ill-posedness of the inverse problem is proved by analyzing the properties of the forward operator. Finally a modified non-stationary iterative Tikhonov regularization method is used to find a stable approximate solution for the potential term. Numerical examples in one- and two-dimensional cases are provided to illustrate the efficiency and robustness of the proposed algorithm.

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.

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.

An inverse space-dependent source problem for a multi-term time fractional diffusion equation

In the present paper, we consider an inverse problem of recovering the space-dependent source for a multi-term time fractional diffusion equation from noisy final data. First, we proved that the direct problem has a unique solution. Second, we proved the existence and uniqueness for the inverse space-dependent source problem. We also prove the ill-posedness of the inverse problem by compactness of input–output mapping. Then, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source. Four different examples are presented to show the feasibility and efficiency of the proposed method.

The positioning of buried pipelines from magnetic data

Buried pipelines are “lifelines” for cities; therefore, it is vital to understand their location and depth before municipal construction to prevent them from being damaged. Magnetic methods have been applied to detect buried ferrous metal pipelines such as steel and cast-iron pipes. We have developed a positioning method for buried pipelines from magnetic data, which is based on a combination of the tilt angle and the downward continuation. The magnetic tilt angle can provide information about the location and depth of buried pipelines, which can easily be calculated by the horizontal and vertical magnetic field gradients. We prove that the tilt angle for the magnetic field that has been reduced to the pole is independent of the magnetization direction given by the pipeline direction with respect to the inducing field. A tilt angle of 90° marks the location of a buried pipeline, whereas the depth is the distance between the location of the 90° and its adjacent 0°. The iterative Tikhonov regularization method for downward continuation, while separating the superimposed anomalies and enhancing the horizontal resolution, also reduces the influence of fast Fourier transform-induced noise and other noise that is intrinsic to the data set. We use the derivative of the Tikhonov regularization result as a regularization term of the minimization function and a constraint for the regularization parameter choice to obtain a more stable and accurate downward-continued result. This positioning method is applicable to single and parallel pipeline detection.

Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation

In this paper, we focus on a backward problem for an inhomogeneous time-fractional diffusion equation on a spherical symmetric domain. This problem is ill-posed in the sense of Hadamard. We propose an iterated fractional Tikhonov regularization method in both cases: the deterministic case and random noise case. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and both rules yield the corresponding convergence rates. The iterated fractional Tikhonov regularization method goes beyond the saturation results of classical Tikhonov method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule. A numerical example is conducted for showing the validity of the proposed method.

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

AbstractThis study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method.

Robust Structural Damage Detection Using Analysis of the CMSE Residual’s Sensitivity to Damage

This paper presents a robust damage identification scheme in which damage is predicted by solving the cross-modal strain energy (CMSE) linear system of equations. This study aims to address the excessive equations issue faced in the assemblage of the CMSE system. A sensitivity index that, to some extent, measures how the actual damage level vector satisfies each CMSE equation, is derived by performing an analysis of the defined residual’s sensitivity to damage. The index can be used to eliminate redundant equations and enhance the robustness of the CMSE system. Moreover, to circumvent a potentially ill-conditioned problem, a previously published iterative Tikhonov regularization method is adopted to solve the CMSE system. Some improvements to this method for determining the iterative regularization parameter and regularization operator are given. The numerical robustness of the proposed damage identification scheme against measurement noise is proved by analyzing a 2-D truss structure. The effects of location and extent of damage on the damage identification results are investigated. Furthermore, the feasibility of the proposed scheme for damage identification is experimentally validated on a beam structure.

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.

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.

An Improved Iterative Tikhonov Regularization Method for Solving the Dynamic Load Identification Problem

The purpose of this study is to identify multi-source dynamic loads acting on the practical engineering structure. In this study, we propose a new iterative Tikhonov regularization method (MITR) based on a new functional operator, and strictly prove its stability and definition from the mathematical theory. This inverse technology has been validated by numerical benchmarks and engineering example involving dynamic load identification. Numerical simulations indicate that the proposed method reduces the number of iterations and also can obtain more efficient and numerically stable approximation of the true loads compared with the iterative Tikhonov regularization method (ITR), which verifies the robustness and effectiveness of the proposed method.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However, downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform (FFT) algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

Observation of a high-energy tail in ion energy distribution in the cylindrical Hall thruster plasma

A novel method is presented to determine populations and ion energy distribution functions (IEDFs) of individual ion species having different charge states in an ion beam from the measured spectrum of an E × B probe. The inversion of the problem is performed by adopting the iterative Tikhonov regularization method with the characteristic matrices obtained from the calculated ion trajectories. In a cylindrical Hall thruster plasma, an excellent agreement is observed between the IEDFs by an E × B probe and those by a retarding potential analyzer. The existence of a high-energy tail in the IEDF is found to be mainly due to singly charged Xe ions, and is interpreted in terms of non-linear ion acceleration.

An adaptive iterative method for downward continuation of potential-field data from a horizontal plane

We have developed an improved adaptive iterative method based on the nonstationary iterative Tikhonov regularization method for performing a downward continuation of the potential-field data from a horizontal plane. Our method uses the Tikhonov regularization result as initial value and has an incremental geometric choice of the regularization parameter. We compared our method with previous methods (Tikhonov regularization, Landweber iteration, and integral-iteration method). The downward-continuation performance of these methods in spatial and wavenumber domains were compared with the aspects of their iterative schemes, filter functions, and downward-continuation operators. Applications to synthetic gravity and real aeromagnetic data showed that our iterative method yields a better downward continuation of the data than other methods. Our method shows fast computation times and a stable convergence. In addition, the [Formula: see text]-curve criterion for choosing the regularization parameter is expressed here in the wavenumber domain and used to speed up computations and to adapt the wavenumber-domain iterative method.

Regularized Iterative Stochastic Approximation Methods for Stochastic Variational Inequality Problems

We consider a Cartesian stochastic variational inequality problem with a monotone map. Monotone stochastic variational inequalities arise naturally, for instance, as the equilibrium conditions of monotone stochastic Nash games over continuous strategy sets or multiuser stochastic optimization problems. We introduce two classes of stochastic approximation methods, each of which requires exactly one projection step at every iteration, and provide convergence analysis for each of them. Of these, the first is a stochastic iterative Tikhonov regularization method which necessitates the update of the regularization parameter after every iteration. The second method is a stochastic iterative proximal-point method, where the centering term is updated after every iteration. The Cartesian structure lends itself to constructing distributed multi-agent extensions and conditions are provided for recovering global convergence in limited coordination variants where agents are allowed to choose their steplength sequences, regularization and centering parameters independently, while meeting a suitable coordination requirement. We apply the proposed class of techniques and their limited coordination versions to a stochastic networked rate allocation problem.

Identification of material parameters of two-dimensional anisotropic bodies using an inverse multi-loading boundary element technique

An inverse technique, based on the boundary element method (BEM) and elastostatic experiments for identification of elastic constants of orthotropic and general anisotropic 2D bodies is presented. Displacement measurements at several points on the boundary of the body, obtained by a few known load cases are used in the inverse analysis to find the unknown elastic constants of the body. Using data from more than one elastostatic experiment results in a more accurate and stable solution for the identification problem. In the inverse analysis, sensitivities of displacements of only boundary points with respect to the elastic constants are needed. Therefore, the BEM is a very useful and effective method for this purpose. An iterative Tikhonov regularization method is used for the inverse analysis. A method for simple computation of initial guesses for unknown elastic constants and a procedure for appropriate selection of the regularization parameter appearing in the inverse analysis is also proposed. Convergence and accuracy of the presented method with respect to measurement errors and number of load cases are investigated by presenting several examples.

Analysis and elimination of the distortion in electro-optic measurement of low-energy subpicosecond electron bunches

Electro-optic (EO) technique is widely used to characterize longitudinal profile of electron bunches. However, electron bunches with a low energy (∼MeV) and a short time duration (∼subpicosecond) cannot be well resolved by EO measurement, which leads to distortion. The convolution theorem is proposed to analyze this distortion in EO measurement. And the factors leading to the distortion are discussed, among which the divergent Coulomb field of electrons is the main one. Distortion elimination and reconstruction of electron bunch profiles from detected EO signals is an ill-posed inverse problem. This paper proposes an iterative Tikhonov regularization method to solve this inverse problem and to reconstruct electron bunch profiles from the EO signals detected by the EO measurement. The feasibility of our proposal is tested and numerically verified based on subpicosecond electron bunches with several MeV energy.

충격응답함수와 조정법을 이용한 항공기 날개의 충격하중 복원 연구

충격응답함수와 조정법(regularization methods)을 이용하여 항공기 날개의 충격하중 복원 가능성을 검토하였다. 충격하중에 대한 구조의 응답을 내타낼 수 있는 충격응답함수를 날개 유한요소모델의 강성과 질량 자료로 유도하였다. 일반적으로 부적합(ill-posed) 특성을 지닌 충격응답함수의 역행렬은 반복 Tikhonov 조종법(Iterative Tikhonov Regularization Method)과 일반화 특이치 분해법(Generalized Singular Value Decomposition Method)을 사용하여 구하였다. 수치적 입증을 위하여 전투기급 주익을 사용하였다. 해당 주익의 유한요소해석을 통하여 임의의 충격하중에 대한 변위와 변형률을 계산하였으며, 이를 충격응답함수로 계산한 결과와 비교하였다. 또한, 유한요소해석에서 계산된 변형률을 사용하여 충격하중을 복원하였다. 수치적 입증 결과 항공기 구조의 충격하중 모니터링이 본 방법으로 가능할 수 있음을 보여주었다. The capability for reconstructing impact forces of an aircraft wing using impact response functions and regularization methods were examined. The impact response function which expresses the relation between the structure response and the impact force was derived using the information on mass and stiffness data of a finite element model for the wing. Iterative Tikhonov regularization method and generalized singular value decomposition method were used to inverse the impact response function that was generally ill-posed. For the numerical verification, a fighter aircraft wing was used. Strain and deflection histories obtained from finite element analysis were compared with the results calculated using impact response functions. And the impact forces were reconstructed with the strain histories obtained from finite element analysis. The numerical verification results showed that this method can be used to monitor impact forces on aircraft structures.