Tikhonov Method Research Articles

A range-relaxed criteria for choosing the Lagrange multipliers in the iterated Tikhonov Kaczmarz method for solving systems of linear ill-posed equations

In this article we propose and analyze a nonstationary iterated Tikhonov Kaczmarz (iTK) type method for obtaining stable approximate solutions to systems of ill-posed equations modeled by linear operators acting between Hilbert spaces. We generalize for the iTK iteration the criteria proposed in [] for the iterated Tikhonov method. The goal is to devise an efficient strategy for choosing the Lagrange multipliers in this method. Convergence analysis for the resulting iTK method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments are presented for two distinct applications, namely: an image deblurring problem and a 2D elliptic parameter identification problem (the inverse potential problem). The obtained numerical results validate the efficiency of the proposed method.

Sparse image reconstruction of intracerebral hemorrhage with electrical impedance tomography.

Purpose: Intracerebral hemorrhage (ICH) is a common disease that is known for its high morbidity, high mortality, and high disability. The fast and accurate detection of ICH is essential for the acute care of patients. Electrical impedance tomography (EIT) offers an alternative with which pathological tissues can be detected by reconstructing conductivity variation. Nevertheless, the sensitive field of EIT is greatly affected by medium distribution, which is referred to as soft-field effect. In addition, the image reconstruction is a severely ill-posed inverse problem. Furthermore, due to the low conductivity of skull, the sensitivity in the sensing area is extremely low. Therefore, the reconstruction of ICH with EIT is great challenge. Approach: A sparse image reconstruction method is proposed for EIT to visualize the conductivity variation caused by ICH. To reduce the impact of soft-field effect, the normalization of sensitivity distribution is conducted for monolayer and three-layer head model. In addition, a constrained sparse -norm minimization model is developed for the image reconstruction. Augmented Lagrangian multiplier method and alternating minimization scheme are adopted to solve the proposed model. Results: The results show that the sensitivity in the sensing area is largely enhanced. Numerical simulation based on monolayer head model and three-layer head model is respectively carried out. Both the reconstructed images and the quantitative evaluations show that image reconstructed by the proposed method is much better than that reconstructed by traditional Tikhonov method. The reconstructions evaluated under the impact of noise also show that the proposed method has superior anti-noise performance. Conclusions: With the proposed method, the quality of the reconstructed image would be greatly improved. It is an effective approach for imaging ICH with EIT technique.

The global navigation systems satellites signals partial powers measuring method

Different consumers of global navigation satellite systems signals are able to receive different combinations of them. For this reason, it is important to monitor the signal parameters separately. The existing partial signals characteristics measuring methods are based on correlation processing, which leads to the loss of measurement information and signal averaging. A method for measuring the partial powers of navigation signals with frequency and code division was developed. The initial data are the total spectral power density measurements results, as well as information on the shape of the partial spectral power densities. A mathematical model of the sum of signals with overlapping spectra was investigated. It was shown that the signal powers were related as coefficients at partial spectral power densities. A system of equations for the coefficients is compiled, its solution is presented by the Tikhonov method. A practical study of the metrological characteristics of the method has been carried out. The research results will be used during monitoring the navigation spacecraft technical condition.

Determination of three parameters in a time-space fractional diffusion equation

In this paper, we consider a nonlinear inverse problem of recovering two fractional orders and a diffusion coefficient in a one-dimensional time-space fractional diffusion equation. The uniqueness of fractional orders and the diffusion coefficient, characterizing slow diffusion, can be obtained from the accessible boundary data. Two computational methods, Tikhonov method and Levenberg-Marquardt method, are proposed to solving this problem. Finally, an example is presented to illustrate the efficiency of the two numerical algorithm.

3-D Image Reconstruction in Planar Array ECT by Combining Depth Estimation and Sparse Representation

In this article, a novel 3-D image reconstruction method was proposed for planar array ECT by combining depth estimation and sparse representation. The central depth of target in the region of interest is first estimated from an image initially reconstructed by the Tikhonov regularization. Second, sparse representation is employed to depict the permittivity distribution using the Gaussian radial basis functions defined at voxels over the plane in the estimated depth. Then, the 3-D image is reconstructed by solving a dedicated optimization problem using the Gauss–Newton method. The proposed method reduces the dimension of the unknowns, improves the condition number of the inverse problem, and enhances the image quality greatly. Numerical simulations and experiments were carried out to validate the proposed method. Results show that the proposed method can provide more accurate 3-D images of single or multiple inclusions at varying depths than conventional Tikhonov, total variation (TV), and Landweber methods while relieving the adverse effect of the nonuniform sensitivity of planar array ECT with varying depths on image reconstruction. The proposed method is of great potential in detecting and imaging subsurface abnormities.

Verification of the RRA-Algorithm Regularization for the Analysis of Stochastic Structures in Bioinformatic Intelligent Systems

This article deals with methods for the analysis of experimental data in intelligent systems for medical monitoring, the control of diagnostic and therapy processes and biomedical facilities. These methods are relevant for creating mathematical models for dynamic systems with stochastic properties as well as for managing complex subsystems of bio information systems (BIS). The nonlinear dynamic properties of such structures and their evolution in terms of random processes and values manifest themselves as complex (non-Gaussian, polymodal) distributions to be subject to correct identification. The distribution law for stochastic characteristics is necessary to form a mathematical probabilistic model of a local object for the BIS control. Based thereon it is possible to present an experimental-mathematical model of the human respiratory or digestive system functioning, to create systems for monitoring of critical parameters in the course of therapy, to predict and synthesize medicines. When identifying the distribution laws for stochastic structures, it is necessary to solve systems of ill-conditioned equations.BIS models in the form of differential or integral equations also require to solve high-order approximating algebraic systems. Algorithms for solving ill-conditioned systems are based on regularization methods. For these purposes the article suggests the RRA-algorithm procedure. The RRA-algorithm is a variation of the Tikhonov method and it is suggested as a regularization method for first kind Fredholm integral equation. To verify the RRA-algorithm for solving systems of algebraic equations with conditioning numbers of level 10 12 and higher, solutions of test first kind Fredholm integral equation are presented.

Effect of Hydrogen Pressure in the Stage of Recombination of HDDR Process on Phase Equilibrium in Nd – Fe – Co – B – Zr-Base Alloys

X-ray diffractometry with mathematical processing of the diffraction patterns by the Tikhonov method of regularization, which enhances the resolution, is used to study the phase state in Nd – Fe – Co – B – Zr-base alloys during the reverse reaction of HDDR process at different hydrogen pressures. The range of hydrogen pressures in which the reverse reaction (recombination) of the HDDR process develops fully is determined experimentally.

Exploring Respiratory Motion Tracking Through Electrical Impedance Tomography

Motion tracking is an effective approach for the management of respiratory motion during the medical imaging process, which has always been a major concern in diagnostic imaging, interventional, and noninvasive therapy. However, the low imaging speed of traditional medical imaging techniques limits the practical application of real-time motion tracking. Electrical impedance tomography (EIT) is proved to be an effective tool for continuous monitoring of lung activity/status. However, the respiratory motion has never been studied in the medical EIT field before. In this article, preliminary research of lung movement during the respiratory process is first studied based on EIT. Multiring electrode thorax models under different respiratory statuses were constructed to obtain simulation data of EIT. A modified TV algorithm is used for the estimation of lung volume and movement based on 3-D EIT images, which improve the quality of reconstruction by approximately 30% and 20% compared with the traditional Tikhonov method and the total variation (TV) method, respectively. Both simulations and experiments were conducted to show the potential of respiratory motion tracking through 3-D EIT reconstruction.

MODERN AREAS OF APPENDIX OF REVERSE TASKS OF MATHEMATICS AND PHYSICS

In this article the questions of modern mathematical design of different processes (computer tomography, non-destructive control, and, exactly, ultrasonic fault detection of and other) come into question from the point of view of theory of reverse tasks. Besides, the article proposes an approach to determining the maximum thermal load measured with a certain error of temperature (thermal) stress by solving the inverse problem of thermal elasticity. To obtain a stable solution of the inverse problem of thermoelasticity, the A.N. Tikhonov method with effective search for the regularization parameter is used. The A.N. Tikhonov functional reflects the deviation of the temperature stress obtained as a result of the observation from the finite element method calculated on the basis of the approximate solution of the direct elasticity problem. Examples of reverse tasks are made, their basic features and prospects of the use are discussed in a design.. Experimentally is theoretical research of thermoelastic descriptions of materials of devices, and also elements of турбостроения comes true coming from the decision of reverse problems of thermoelasticity. Greate importance in solving inverse problems are the issues of measurement uncertainty, which affects the final results.

Electrocardiographic imaging including intracardiac information to achieve accurate global mapping during atrial fibrillation

Atrial fibrillation (AF) is characterized by complex and irregular propagation patterns. Noninvasive electrocardiographic imaging (ECGI) has been tested during AF conditions with promising results. However, current regularization methods face important challenges in this type of unstable electrical activity scenarios. Combination of intracardiac and non-invasive simultaneous recordings could improve ECGI performance and allow real-time global mapping of complex AF patterns. In this work, we propose an ECGI method that incorporates intracardiac measurements as a constraint in a reformulation of the classical Tikhonov method. We used realistic mathematical models of atria and torso that simulates a wide number of epicardial electrical activity patterns. Body surface potentials were obtained from simulated electrograms (EGMs) by using Boundary Element Method and corrupted with Gaussian noise. Epicardial potentials were estimated using inverse problem with Tikhonov regularization, including intracavitary information as a second constraint. Results showed that first-order Constrained Tikhonov formulation provided more reliable reconstructions than the classical Tikhonov approach in AF conditions using at least 32 uniformly distributed endocardial EGMs (CC between 0.87 and 0.28, depending on the AF complexity). Constrained Tikhonov provided more accurate spatial mass functions (SMF) of PS locations (CCSMF between 0.24 and 0.86). This methodology was tested on real patient data, obtaining a mean DF RMSE of 0.85 Hz, outperforming the classical Tikhonov approach. Limitations of this study include the fact that the model considered endocardium and epicardium as a single layer. Further research will include endocardium–epicardium bilayer model approximations and validation using more real patient data.

Optimal combinations of Tikhonov regularization orders for IHCPs

In achieving a regularized solution of inverse heat conduction problems (IHCPs), Tikhonov Regularization works based on adding either a zeroth-, a first-, or a second-order term to the sum of squared errors function. In other words, considering combinations of these terms are not often considered. This work investigates employing standard optimization techniques in order to obtain optimal regularization parameters when combinations of these three regularization terms are used. Five different heat pulses are used as test cases: step, triangular, quadratic, quartic, and half-sine. The criterion used to find the optimal value for the regularization parameters is sum of the squares of deviations between the estimated heat flux and the exact heat flux pulse. A hybrid method which utilizes both the Genetic Algorithm and the Pattern Search is used for the optimization through functions defined in MATLAB software. Moreover, a general case containing all five heat flux test cases is considered in finding the optimal combination of three versions of Tikhonov method. All separate and general optimal combined models are used in estimating five pulse heat flux functions, and for each case, the RMS errors are calculated to give an insight toward the combined Tikhonov regularization technique. The proposed approach is also used to recover the surface heat flux using measured temperature data from an experiment. Some discussion about the Morozov discrepancy principle for this application is considered at the end.

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.

An inexact non stationary Tikhonov procedure for large-scale nonlinear ill-posed problems

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing a Lanczos approach, by which an approximated solution is computed. The ad-hoc choice of the trust region radius update and the structure of the step resulting from the use of the Lanczos approach, allows us to prove some regularizing properties of the method. The proposed approach is tested on a parameter identification problem and on an image registration problem, and it is shown to provide important computational savings with respect to its exact counterpart.

Comparison of unfolding methods using RooFitUnfold

In this paper we describe RooFitUnfold, an extension of the RooFit statistical software package to treat unfolding problems, and which includes most of the unfolding methods that commonly used in particle physics. The package provides a common interface to these algorithms as well as common uniform methods to evaluate their performance in terms of bias, variance and coverage. In this paper we exploit this common interface of RooFitUnfold to compare the performance of unfolding with the Richardson–Lucy, Iterative Dynamically Stabilized, Tikhonov, Gaussian Process, bin-by-bin and inversion methods on several example problems.

Solving the Riesz\u2013Feller space-fractional backward diffusion problem by a generalized Tikhonov method

The article investigates a Riesz–Feller space-fractional backward diffusion problem. We develop a generalized Tikhonov regularization method to overcome the ill-posedness of this problem, and then based on the result of conditional stability, we derive the convergence estimates of logarithmic and double logarithmic types for the regularized method by adopting a-posteriori choice rules of regularization parameter. Finally, by using the finite difference method, we solve a direct problem to construct the data, and some corresponding results of numerical simulations are presented to verify the convergence and stability for this method.

Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization

We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level.

TSVD and Tikhonov methods and influence factor analysis for NMR data in shale rock

Nuclear magnetic resonance (NMR) technique is a powerful tool in both conventional and unconventional reservoirs. While signal decays fast in shale reservoirs/rocks and signal-to-noise ratio (SNR) is low. It is necessary to study inversion methods and regularization algorithms which are suited to shale rocks. This work studied two inversion methods (TSVD and Tikhonov) and their related regularization algorithms (BGL, Morozov, GCV and L-curve). Numerical simulations and rock analysis indicated both methods are suited to shale rocks. Especially, Tikhonov is better than TSVD in case SNR is low.We also studied the influences of NMR measurement parameters on the inversion, including SNR, number of T2 bins, number of echoes and echo space TE. The numerical simulation and rock analysis are consistent with the theoretical results. Furtherly, we advise the extension into NMR logging in shale reservoirs, it should meet the conditions: SNR ≥5, number of T2 bins ≥16, number of echoes ≥50, and TE ≤ 0. 2 ms. It is possible to acquire fewer echoes in shale reservoirs, which can make logging speed up.

Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

In this paper, we consider an inverse problem of identifying the source term for a generalization of the time-fractional diffusion equation, where regularized hyper-Bessel operator is used instead of the time derivative. First, we investigate the existence of our source term; the conditional stability for the inverse source problem is also investigated. Then, we show that the backward problem is ill-posed; the fractional Landweber method and the fractional Tikhonov method are used to deal with this inverse problem, and the regularized solution is also obtained. We present convergence rates for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Finally, we present a numerical example to illustrate the proposed method.

Spatial-dependent regularization to solve the inverse problem in electromyometrial imaging.

Recently, electromyometrial imaging (EMMI) was developed to non-invasively image uterine contractions in three dimensions. EMMI collects body surface electromyography (EMG) measurements and uses patient-specific body-uterus geometry generated from magnetic resonance images to reconstruct uterine electrical activity. Currently, EMMI uses the zero-order Tikhonov method with mean composite residual and smoothing operator (CRESO) to stabilize the underlying ill-posed inverse computation. However, this method is empirical and implements a global regularization parameter over all uterine sites, which is sub-optimal for EMMI given the severe eccentricity of body-uterus geometry. To address this limitation, we developed a spatial-dependent (SP) regularization method that considers both body-uterus eccentricity and EMG noise. We used electrical signals simulated with spherical and realistic geometry models to compare the reconstruction accuracy of the SP method to those of the CRESO and the L-Curve methods. The SP method reconstructed electrograms and potential maps more accurately than the other methods, especially in cases of high eccentricity and noise contamination. Thus, the SP method should facilitate clinical use of EMMI and can be used to improve the accuracy of other electrical imaging modalities, such as Electrocardiographic Imaging. Graphical abstract The spatial-dependent regularization (SP) technique was designed to improve the accuracy of Electromyometrial Imaging (EMMI). The top panel shows the eccentricity of body-uterus geometry and four representative body surface electrograms. The bottom panel shows boxplots of correlation coefficients and relative errors for the electrograms reconstructed with SP and two conventional methods, the L-Curve and mean CRESO methods.

The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation

In this paper, we identify the initial value for a time-fractional diffusion equation on a columnar axis-symmetric domain. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the domain. Two different kinds of fractional Tikhonov methods are used to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solutions and the exact solution are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our methods.