Tikhonov Method Research Articles

A Tikhonov regularization method for solving a backward time–space fractional diffusion problem

In this paper, a backward problem for a time–space fractional diffusion equation is considered, which is to determine the initial data from a noisy final data. To deal with this ill-posed problem, combined the ideas of the Tikhonov regularization in Hilbert Schales proposed by Natterer in 1984 and the fractional Tikhonov method given by Hochstenbach–Reichel in 2011, a Tikhonov regularization method is constructed. Based on the Fourier transform, an a-priori and an a-posteriori regularization parameter choice rules are used to guarantee the order optimal convergence rates. By the finite difference methods and the Discrete Fourier transform, numerical examples in one-dimensional and two-dimensional cases are given for the a-posteriori regularization parameter choice rule. Theoretical and numerical results show that the proposed method works well for both the smooth and the non-smooth functions.

A wavelet frame constrained total generalized variation model for imaging conductivity distribution

<p style='text-indent:20px;'>Electrical impedance tomography (EIT) is a sensing technique with which conductivity distribution can be reconstructed. It should be mentioned that the reconstruction is a highly ill-posed inverse problem. Currently, the regularization method has been an effective approach to deal with this problem. Especially, total variation regularization method is advantageous over Tikhonov method as the edge information can be well preserved. Nevertheless, the reconstructed image shows severe staircase effect. In this work, to enhance the quality of reconstruction, a novel hybrid regularization model which combines a total generalized variation method with a wavelet frame approach (TGV-WF) is proposed. An efficient mean doubly augmented Lagrangian algorithm has been developed to solve the TGV-WF model. To demonstrate the effectiveness of the proposed method, numerical simulation and experimental validation are conducted for imaging conductivity distribution. Furthermore, some comparisons are made with typical regularization methods. From the results, it can be found that the proposed method shows better performance in the reconstruction since the edge of the inclusion can be well preserved and the staircase effect is effectively relieved.</p>

Determine unknown source problem for time fractional pseudo-parabolic equation with Atangana-Baleanu Caputo fractional derivative

<abstract><p>In this paper, we consider a pseudo-parabolic equation with the Atangana-Baleanu Caputo fractional derivative. Our main tool here is using fundamental tools, namely the Fractional Tikhonov method and the generalized Tikhonov method, the error estimate is shown. Finally, we provided numerical experiments to prove the correctness of our theory.</p></abstract>

An Adaptive Regularized Solution to Inverse Ill-Posed Models

The ill-posed models are widely encountered in various inversions of geodesy and remote sensing. The regularization approaches can significantly stabilize the solution to ill-posed models since the high-frequency noise is effectively suppressed. Although the famous Tikhonov regularization and truncated singular value decomposition (TSVD) regularization have been widely applied in various geodetic applications, there still remain theoretical drawbacks for either single regularization. For Tikhonov regularization, given a regularization parameter, the low-frequency terms are over regularized, and high-frequency terms are under regularized. For TSVD regularization, some medium-frequency terms will be mistaken for high-frequency terms to be truncated and the hidden signals will be lost. For this reason, we propose an adaptive regularized solution in spectral form, which adaptively divides the terms of different frequencies into three kinds: (i) the low-frequency terms are not regularized; (ii) the medium-frequency terms are regularized by the Tikhonov method; (iii) the high-frequency terms are regularized by TSVD method. The analytical conditions for determining the term sets are derived based on the criteria that the introduced biases should be smaller than the reduced errors, in other words, the mean squared error (MSE) should be reduced. The two examples are presented to demonstrate the performance of our adaptive regularization. The first numerical example is solving the Fredholm integral equation of the first kind, which is widely encountered in remote sensing inversions. The simulations clearly demonstrate that the adaptive regularized solution can improve the MSE of ordinary Tikhonov and TSVD regularized functions by 25.00% and 9.09%, respectively; In the second example, we apply the new method to investigate the mass variation of the Yangtze River Basin based on the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field model. The Tongji-Grace 2018 monthly gravity field solutions from April 2002 to December 2016 are used to construct the mascon observation equation. The results show that our method also outperforms the ordinary Tikhonov and TSVD regularized solutions, with mean MSE reductions of about 13.40% and 11.69%, respectively. Furthermore, the spatial resolution of secular trend derived by our method are improved and the signal-to-noise ratio (SNR) of mass variation series is higher than the other two regularizations.

A quasi-boundary method for solving an inverse diffraction problem

<abstract><p>In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary $ z = 0 $, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.</p></abstract>

Tensor Arnoldi–Tikhonov and GMRES-Type Methods for Ill-Posed Problems with a t-Product Structure

This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in (Linear Algebra Appl 435:641–658, 2011). A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.

Determination Transition Resistance of Cathode-Polarized Main Pipeline on Magnetometery Data

The article deals with the inverse problem of determining the transient resistance of the main pipeline insulating coating. For this, UAV measurements of the magnetic induction vector modulus of the magnetic field excited by the system of electrochemical cathodic protection of pipelines are used. The solution method is based on Tikhonov's method for finding the extremal of the regularizing functional. The developed algorithm is implemented in software. The results of computational experiments are presented.

Reconstruction algorithm for impenetrable rough surface profile under Neumann boundary condition

In this paper, an algorithm to reconstruct one-dimensional impenetrable rough surface from the knowledge of scattering field is presented. The rough surface is considered as locally perturbed and the scattering field data are collected above the roughness in a simple non-magnetic medium considering Neumann boundary condition. First, the surface integral equation constituted via the Neumann boundary condition is solved and scattering field data are observed synthetically. Then, the same surface integral equation together with the data equation are solved in an iterative fashion to reconstruct the surface variation. In the numerical implementation, the so-called ill-posed inverse problem is regularized with Tikhonov method and a least-squares solution is obtained by using Gaussian-type basis function. Finally, numerical examples are carried out to illustrate effectiveness of the method.

Total fractional-order variation regularization based image reconstruction method for capacitively coupled electrical resistance tomography

Compared with electrical resistance tomography, capacitively coupled electrical resistance tomography (CCERT) is preferred since it avoids problems of electrode corrosion and electrode polarization. However, reconstruction of conductivity distribution is still a great challenge for CCERT. To improve reconstruction quality, this work proposes a novel image reconstruction method based on total fractional-order variation regularization. Simulation work is conducted and reconstruction of several typical models is studied. Robustness of the proposed method to noise is also conducted. Additionally, the performance of the proposed reconstruction method is quantitatively evaluated. We have also carried out phantom experiment to further verify the effectiveness of the proposed method. The results demonstrate that the quality of reconstruction has been largely improved when compared with the images reconstructed by Landweber, Newton-Raphson and Tikhonov methods. The inclusion is more accurately reconstructed and the background is much clearer even under the impact of noise.

The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

AbstractThis paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large‐scale linear discrete ill‐posed problems based on the t‐product formalism for third‐order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641‐658). The solution methods presented first reduce a given (large‐scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third‐order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third‐order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill‐posed problems defined by a matrix operator to linear discrete ill‐posed problems defined by a third‐order tensor operator. An interlacing property of singular tubes for third‐order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t‐product approach, in comparison with solution methods that are based on matricization of the tensor equation.

Reconstructing the right-hand side of the Rayleigh\u2013Stokes problem with nonlocal in time condition

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.

On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

Application of Tikhonov regularization to reconstruct past climate record from borehole temperature

We present a numerical Tikhonov regularization method that can be used to reconstruct past ground surface temperature (GST) records from borehole temperatures. The present ground temperature preserves past climatic signals according to the heat diffusion process in permafrost-affected soils. To track past surface temperature, we employ an inverse method based on a physical connection between GST and measured borehole temperatures. We validate this method by applying it to two synthetic surface temperature cases. Since measured borehole data include uncertainty, we add random noise to our synthetic input borehole data to simulate the process of noise suppression. GST recovered with corresponding uncertainty shows a close match with synthetic surface temperature for both cases. We show that this method can successfully suppress the noise disturbance and achieve smoother solutions.The ability of borehole temperature data to resolve past climatic events is investigated using the Tikhonov method. We investigated past GST of Wudaoliang on Qinghai-Tibet plateau and the inversion result shows the increasing trend of 1.8 (±1.6) in the past 308 years. This GST trend fits the air temperature observation trend but has small value deviation caused by local topography and surface energy budgets of the ground surface.

Estimation of electron fluxes in the Earth's geostationary orbit with Tikhonov regularization during a magnetically quiet period

Electron energy fluxes in the Earth's outer radiation belt are estimated using an inverse theory of the Tikhonov regularization based upon observations in geostationary orbits. Particle Detector (PD) experiment aboard a geostationary satellite GEO-KOMPSAT-2A (GK2A) at a geographic longitude of 128.2°E provided observations of electrons within a 150–2,400 keV energy range with an unprecedented energy resolution of ΔE/E in the range of 5–25%. Instrument response functions, calculated with Monte-Carlo simulations, are deconvoluted with electron observations. Using regularization parameters determined from Generalized Cross Validation (GCV), the Tikhonov method was applied to observations made during a geomagnetically quiet period. This Tikhonov regularization method, now possible for observations in the Earth's radiation belt for the first time, allows direct inference of electron fluxes without resorting to predetermined functional forms. Comparisons of our results with those from conventional methods indicate differences among the results as large as ∼200%.

Optimal Regularization Method Based on the L-Curve for Solving Rational Function Model Parameters

Rational polynomial coefficients in a rational function model (<small>RFM</small>) have high correlation and redundancy, especially in high-order <small>RFMs</small>, which results in ill-posed problems of the normal equation. For this reason, this article presents an optimal regularization method with the L-curve for solving rational polynomial coefficients. This method estimates the rational polynomial coefficients of an <small>RFM</small> using the L-curve and finds the optimal regularization parameter with the minimum mean square error, then solves the parameters of the <small>RFM</small> by the Tikhonov method based on the optimal regularization parameter. The proposed method is validated in both terrain-dependent and terrain-independent cases using Gaofen-1 and aerial images, respectively, and compared with the least-squares method, L-curve method, and generalized cross-validation method. The experimental results demonstrate that the proposed method can solve the <small> RFM</small> parameters effectively, and their accuracy is increased by more than 85% on average relative to the other methods.

A Deep Neural Network Method for Arterial Blood Flow Profile Reconstruction.

Arterial stenosis will reduce the blood flow to various organs or tissues, causing cardiovascular diseases. Although there are mature diagnostic techniques in clinical practice, they are not suitable for early cardiovascular disease prediction and monitoring due to their high cost and complex operation. In this paper, we studied the electromagnetic effect of arterial blood flow and proposed a method based on the deep neural network for arterial blood flow profile reconstruction. The potential difference and weight matrix are used as inputs to the method, and its output is an estimate of the internal blood flow velocity distribution for arterial blood flow profile reconstruction. Firstly, the weight matrix is input into the convolutional auto-encode (CAE) network to extract its features. Then, the weight matrix features and potential difference are combined to obtain the features of the blood velocity distribution. Finally, the velocity features are reconstructed into blood flow velocity distribution by a convolution neural network (CNN). All data sets are obtained from a model of the carotid artery with different rates of stenosis in a uniform magnetic field by COMSOL. The results show that the average root mean square error of the reconstruction results obtained by the proposed method is 0.0333, and the average correlation coefficient is 0.9721, which is better than the corresponding indicators of the Tikhonov, back propagation (BP) and CNN methods. The simulation results show that the proposed method can achieve high accuracy in blood flow profile reconstruction and is of great significance for the early diagnosis of arterial stenosis and other vessel diseases.

On the stabilization and extension of the distribution of relaxation times analysis

The Distribution of Relaxation Times (DRT) is a useful technique to provide an improved insight into the interpretation of Electrochemical Impedance Spectroscopy (EIS) data. Despite its capability to distinguish the characteristic frequencies of the processes involved in an electrochemical system, the DRT solution is an ill-posed problem that requires a regularization method. Tikhonov regularization is one of the most commonly adopted methods for DRT analysis. Nevertheless, the domain discretization, needed to solve the original continuous problem, introduces errors in DRT evaluation. This problem is quite relevant when the impedance is not sufficiently resolved at boundary frequency range (imaginary part of impedance does not approach 0), affecting the edges of the DRT spectrum and the internal part. Here, the zero-padding technique is applied to overcome this issue. The results show how this approach, frequently used in the signal theory but not in DRT, improves DRT evaluation quality with Tikhonov regularization. It consists of an extension of the experimental frequency domain, reducing the errors due to the frequency truncation. The extended algorithm, named ED-DRT, is compared with the conventional Tikhonov method on both simulated and real impedance data. The new approach increases the quality of DRT results, providing higher stability and accuracy in evaluating characteristic frequencies and global polarization resistance even with noisy EIS data.

THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS

We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.

REGINN-IT method with general convex penalty terms for nonlinear inverse problems

In the spirit of the method in Margotti and Rieder (An inexact Newton regularization in Banach spaces based on the non-stationary iterated Tikhonov method. J Inverse Ill-Posed Probl. 2015;23:373–392), we propose an inexact Newton regularization which is based on the non-stationary iterated Tikhonov method for non-smooth solutions of nonlinear inverse problems in Banach spaces. The method consists of an inner iteration and an outer iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method. The inner iteration is performed by the non-stationary iterated Tikhonov method. The remarkable point is that the penalty terms can be general convex functions. Under certain assumptions on nonlinear operators, the convergence analysis of our method is given by making use of tools from convex analysis. Furthermore, numerical simulations are provided to support the theoretical results and test the performance of the method.

Recovering the source term for parabolic equation with nonlocal integral condition

The main purpose of this article is to present a Tikhonov method to construct the source function f(x) of the parabolic diffusion equation. This problem is well known to be severely ill‐posed. Therefore, regularization is required. The error estimates between the sought solution and the regularized solution are obtained under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. One numerical test illustrates that the proposed method is feasible and effective.