Tikhonov Regularization Method Research Articles

Stable identification of sources located on the cerebral cortex from EEG over the scalp

In this work, we present a stable algorithm for the inverse problem of identifying cortical sources from electroencephalographic measurements on the scalp. This inverse problem is ill-posed due to the numerical instability it presents, i.e., small changes in the measurements can produce large variations in the location of the sources. A boundary value problem is used to find correlations between the sources and measurements. In the case in which the head is modeled using two concentric spheres. We use spherical harmonics to find the solution to the inverse source cortical problem. To handle the numerical instability of this problem, we use the Tikhonov regularization method and a cut-off of the harmonic expansion series. From numerical tests, we found these parameters with which we get good approximations. Finally, we illustrate the algorithm proposed with synthetic examples

Asymptotic method and transient terms in exact controls

There is a narrow but hidden link between optimal control theory and the so-called Tikhonov regularization method. In fact, the small coefficient representing the marginal cost of the control can be interpreted as the regularization parameter in a Tikhonov method as far as there exists an exact control. This strategy enables one to adjust the cost function in the optimal control model in order to define the exact control which minimizes a given functional involving both the control but also the state variables during the control process. The goal of this paper is to suggest a method which gives a simple way to characterize and compute the exact control corresponding to the minimum of a given cost functional as said above. It appears as an extension of the phase control which is a finite dimensional version of the HUM control of J.L. Lions but for partial differential equations.

Semi-discrete Tikhonov regularization in RKHS with large randomly distributed noise

The concern of this paper is the numerical solution of the moderately ill-posed first kind integral equations in reproducing kernel Hilbert spaces. Different with previous works in (Krebs et al 2009 J. Inverse Ill-Posed Problems 17 845–69; Zhong et al 2012 Inverse Problems 28 065019), the amount of observation data is large, and corrupted by randomly distributed noise with large variance. In this paper, a purely data driven semi-discrete Tikhonov regularization method is proposed, effectively reconstructing the sought solution. The choice rules of parameters are provided and the rigorous upper bound estimation of confidence interval of the error in L 2 norm is established. Some numerical examples are provided to illustrate the appropriateness of the parameter choice and show the computational performances of the method.

Determining an unknown source in a time-fractional diffusion equation based on Jacobi polynomials expansion with a modified Tiknonov regularization

In this paper, we try to recover an unknown source in a time-fractional diffusion equation. In order to overcome the influence of boundary conditions on source conditions, we introduce the Jacobi polynomials to construct the approximation and a modified Tikhonov regularization method is proposed to deal with the illposedness. Error estimates are obtained under a discrepancy principle as the parameter choice rule. Numerical results are also presented to demonstrate the effectiveness of the proposed method.

A Multipath Error Reduction Method for BDS Using Tikhonov Regularization with Parameter Optimization

Multipath error is an important factor restricting the relative positioning accuracy of the Beidou Navigation Satellite System (BDS). Because of the complexity of the reflection environment, the mathematical modeling of multipath errors is quite difficult. The sidereal filtering algorithm corrects multipath errors by using the feature of period repetition, which can greatly reduce its influence and improve the accuracy of positioning and attitude measurement. In view of the constellation heterogeneity of BDS, it is more complicated to apply sideral filtering. Based on the reconstructed single-difference residual of the carrier phase, the multipath repetition time of the Beidou satellite is estimated using the idea of segmentation. The Tikhonov regularization method and the classical wavelet method are used to extract the multipath of the single-difference residual of the carrier phase, and the “clean” sequence of the single-difference residual is obtained. The experimental results show that it is feasible to extract the multipath error correctly by Tikhonov regularization, and the multipath error is smoother than the original residual measurement. Furthermore, the estimation method of the regularization parameter is further optimized. After using the optimized Tikhonov regularization method with sidereal filtering, the mean RMS improvements of GEO, IGSO, and MEO satellites are 45.9%, 38.2%, and 37.5%, respectively. The positioning accuracy on E, N, and U components is improved by 24.8%, 26.3%, and 42.7%, respectively. The attitude resolution accuracy is improved by 22.9% in the yaw angle and 12.6% in the pitch angle. The proposed method can be an alternative BDS multipath error modeling and mitigation approach.

APPLICATION OF SPLINE FUNCTIONS AND WALSH FUNCTIONS IN PROBLEMS OF PARAMETRIC IDENTIFICATION OF LINEAR NONSTATIONARY SYSTEMS

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task.
 Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions.
 Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval.
 Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters.
 Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters.

About the Tikhonov Regularization Method for the Solution of Incorrect Problems

From time to time, papers are published containing gross errors when solving integral equations of the first kind. This paper is devoted to the analysis of these errors. The paper considers Tikhonov’s weak and operator regularization. To construct a solution to the integral equation, the local splines of the Lagrangian type of the second order of approximation, as well as the local splines of the Hermitian type of the fourth order of approximation of the first level, are used. The results of numerical experiments are presented.

A Tikhonov regularization method for Cauchy problem based on a new relaxation model

In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.

Load Identification of High-Speed Train Crossbeam Based on Bayesian Finite Element Model Updating and Load-Strain Linear Superposition Algorithm

The load identification of the crossbeams in high-speed trains is proving to be crucial, because of the complexity of its structure and environment. In this paper, a method to identify load of the crossbeam structure based on Bayesian finite element model updating and load-strain linear superposition algorithm is proposed. Firstly, the finite element model is established. Using the numerical simulation under different loads, the relative mean error of load identification is less than 0.5% and the effect of strain field reconstruction performs well. Based on the model test system, Markov Chain Monte Carlo method is used to update the parameters of the initial finite element model. Then an experimental setup is built with 64 FBG sensors installed on the crossbeam. The load identification results of Tikhonov regularization method, the load-strain linear superposition method before and after the Bayesian finite element model updating are analyzed and compared. The results show that the proposed method can significantly improve the accuracy of load identification.

Two regularization methods for identifying the source term of Caputo–Hadamard time‐fractional diffusion equation

In this paper, the inverse problem for identifying the unknown source on time‐fractional diffusion equation with Caputo–Hadamard derivative is considered. This problem is ill‐posed, and two regularization methods are used to solve it. Firstly, we prove that the problem is ill‐posed and give the conditional stability and the optimal error bound. Secondly, the error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under a priori and a posteriori parameter selection rules are given, respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.

An improved homotopy perturbation method for dynamic force reconstruction

In many engineering application, it is still a challenge in directly obtaining external force acting on structures. Moreover, many researchers attach more attention to indirect measurement technology using some response signals. This paper presents an improved homotopy perturbation method (IHPM) for dynamic load identification. This method can identify multi-source dynamic sine and triangle force. Firstly, the present method is proposed based on improved homotopy operator. Secondly, the sequence that the present method generates is proved to be a Cauchy sequence when ‖Aeα−I‖<1. Finally, the proposed method is applied to the dynamic load identification of hanging basket structure. Numerical simulations verify the stability and effectiveness of the newly developed method, and also validate that it is superior to the traditional homotopy perturbation method (HPM) and Tikhonov regularization method.

Compressive sensing-based tomography for Absolute X-ray UltraViolet (AXUV) diagnostics.

Absolute x-ray ultraviolet diagnostics ensures 2D coverage of the radiation emission region that enables tomographic reconstruction. However, retrieving the local emissivity via tomography remains a challenge due to its ill-posed nature. Tikhonov regularization with smoothness operation generally performs well but tends to over-smooth regions with steep gradients and local structure in the radiation profile and may introduce artifacts. In this paper, a tomography method based on compressive sensing theory with Tikhonov regularization terms is developed. Experimental results on multiple phantom sets show that the proposed method improves the reconstruction accuracy and quality in regions with steep gradients compared with the Tikhonov regularization method and suppresses the unphysical negative emissivity. The analysis of reconstruction uncertainty shows that the dictionary learning process provides more accurate prior information about steep gradients to improve the quality of reconstructed images, and compressive sensing has the denoising capability to reduce the impact of noise. Finally, the method is validated by data from the Sino-UNIted Spherical Tokamak, showing fewer artifacts and more reliable reconstruction images than the earlier method.

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh, Appl. Math. Comput. 392, 125701, 2021] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

Fast multilevel iteration methods for solving nonlinear ill-posed problems

Abstract We propose a multilevel iteration method for the numerical solution of nonlinear ill-posed problems in the Hilbert space by using the Tikhonov regularization method. This leads to fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. An adaptive choice of an a posteriori rule is suggested to choose the stopping index of iteration, and the rates of convergence are also derived. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

On the Estimation of the Sensitivity of the Coefficients of Models for Inhomogeneous Solids

The problem of determining the characteristics of the sensitivity of inhomogeneous isotropic elastic solids with respect to three functions characterizing the inhomogeneity - Lame coefficients and density - is considered. The corresponding boundary value problems are formulated, equations for determining the sensitivity are obtained. An example is presented for a cylindrical rod in the analysis of longitudinal and bending oscillations, the sensitivity to changes in Young's modulus and density is analyzed. Using the iterative algorithm and the A.N. Tikhonov's regularization method, a number of inverse problems were solved to determine these characteristics for monotone and non-monotone laws of their change, and the results of computational experiments are presented. The norm of the difference between the exact and reconstructed solutions is analyzed. Recommendations are given for choosing the most efficient frequency range for sounding from the point of view of reconstruction.

Regularization Methods for Extracting Doppler Broadening From Hall Thruster Plasma

Extracting Doppler broadening from laser-induced fluorescence (LIF) measurements is essential for determining species temperature in the plasma. The fluorescence signal is a convolution of many broadening effects of which Doppler broadening is the most significant, but several others contribute as well: collisional (pressure) broadening (particle-to-particle interactions), natural lifetime broadening, and power broadening. The process that extracts Doppler broadening is an <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ill-posed</b> inverse problem. The regularization approach addresses the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ill-posed</b> problem to ensure a unique and stable solution with the best possible accuracy. This study reported the Xe II temperature by applying truncated singular value decomposition (TSVD) and Tikhonov regularization (zeroth, first and second-order) methods for deconvolution of Doppler broadening. Tikhonov first-order and TSVD employing the generalized cross-validation (GCV) method provide accurate reconstruction within 2.0%–8.0% of the actual measurements.

Inverse identification model for release rates of multiple gaseous pollution sources in an aircraft cabin

When gaseous pollutants are airborne in an aircraft cabin, it is important to know the release rate of pollution sources efficiently and accurately for effective mitigation. At present, it is possible to locate single or multiple pollution sources using the sensor information, but there is no fast and accurate method to determine the strength of multiple pollution sources. In this study, using the monitored pollutant concentrations, with the assumption that the location of multiple pollution sources in an aircraft cabin is known, an inverse model for determining the release rates was established. In order to improve the calculation efficiency, the cause-effect matrix between the release rates of pollution sources and the concentrations of monitored points was obtained by introducing the contribution ratio of pollutant sources (CRPS) method. Due to the direct inversion of the cause-effect matrix being ill-posed, the Tikhonov regularization method was used to enhance the stability of the inverse solution. The inverse model was validated by experiment in a three-dimensional cavity with CO2 as a tracer gas. The method was further demonstrated in a three-dimensional aircraft cabin by simulated data. The results show that the inverse modelling can accurately and efficiently quantify release rates of multiple pollution sources.

Imaging of intracerebral hemorrhage with adaptive genetic algorithm in brain electrical impedance tomography

Intracerebral hemorrhage refers to bleeding caused by the spontaneous rupture of blood vessels. Accurate diagnosis of hemorrhage is vital in the treatment of a patient. As a new medical imaging technique, electrical impedance tomography (EIT) is able to offer images of conductivity distribution variation caused by pathological change. However, image reconstruction of EIT suffers from the problem of serious ill-posedness. In particular, in brain imaging, irregular and multi-layered head structure together with the low conductivity of the skull further aggravate the problem. In order to address this problem, a new image-reconstruction method is proposed for imaging of hemorrhage in this work. With current solutions solving by a Tikhonov regularization method for the original conductivity distribution, the proposed method enhances the reconstruction quality by introducing an adaptive genetic algorithm. To test the performance of the proposed method, simulation work is conducted. A three-layer head model is established and an inclusion, which simulates hemorrhage, is placed at six different locations in the brain layer. Images reconstructed by the Tikhonov method, Newton–Raphson method and the traditional genetic algorithm are used for comparisons. Quantitative evaluation is also performed. The anti-noise performance of the proposed method is estimated by considering noise with differing signal-to-noise ratios. In addition to simulation, phantom experiments are carried out to further verify the performance of the proposed method. The results show that the proposed method performs well in the reconstruction of simulated intracerebral hemorrhage. With the proposed method, the inclusion can be more accurately reconstructed and the background is much clearer than the other three traditional methods.

A probabilistic method for dynamic force identification of uncertain structures

To address the dynamic force identification problem of uncertain structures with random parameters, this study proposes a novel approach to describe the probabilistic properties of the identified forces. First, the integral equation for the force probability density function (PDF) is derived from the mathematical mapping relationship between the identified force and random structural parameters. Next, the sample space of the random parameters is partitioned into non-overlapping subdomains using a point selection technique based on the generalized F-discrepancy. Meanwhile, the function fitting technique and Tikhonov regularization method are applied to identify the force samples under the corresponding partitioned subdomains. Thus, the integral equation of the force PDF is transformed into a summation of the force probabilities for all the subdomains. Subsequently, a smoothing technique is introduced to obtain the expression of the identified force PDF. Finally, the confidence interval over the entire time domain is constructed from the force cumulative distribution function, which effectively quantifies the influence of random structural parameters on the identified results. The efficiency and superiority of the proposed method are verified using an example of a plane truss structure with multiple random structural parameters, and force identification for a simplified launch vehicle model with connection uncertainties is discussed.

Reconstruction of conductivity distribution variation with Tikhonov and wavelet frame combined method for electrical impedance tomography

Electrical impedance tomography (EIT) is receiving considerable research interest in a variety of applications. With this technique, it is possible to reconstruct conductivity distribution inside the detected region. Note that reconstruction of conductivity distribution is an ill-posed inverse problem. To deal with this problem, Tikhonov regularization method has been preferred. However, this method suffers from poor reconstruction quality. In this work, a novel approach which combines Tikhonov regularization method with wavelet frame is proposed for image reconstruction of EIT. The proposed method is superior to the popular Tikhonov regularization method in enforcing sparse of the solution and enhancing sharp feature. Simulation work has been conducted to validate the performance of the proposed method. Compared with Landweber method, total variation (TV) regularization method, and Tikhonov regularization method, reconstructions of six different models show that images reconstructed by the proposed method have better quality and are more robust to noise. In addition, image reconstruction is also performed based on phantom experimental data. The results further demonstrate that the proposed method outperforms other three methods since the inclusion has been more accurately reconstructed, and there are almost no artifacts in the reconstructed images.