Iterated Tikhonov Regularization Research Articles

An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems

With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems 39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

An iterative Tikhonov regularization downward continuation of magnetic anomaly

An iterative Tikhonov regularization downward continuation of magnetic anomaly

Regularized Nyström Subsampling in Covariate Shift Domain Adaptation Problems

The unsupervised domain adaptation problem with covariate shift assumption is considered. Within the framework of the Reproducing Kernel Hilbert Space concept, an algorithm is constructed that is a combination of the Nyström subsampling and the iterated Tikhonov regularization. This approach allows significantly reduce the amount of computing resources involved and at the same time achieves the minimal (by order) approximation accuracy under the big data settings.

Nonstationary iterated frozen Tikhonov regularization with uniformly convex penalty terms for solving inverse problems

In this paper, we propose the frozen form of nonstationary iterated Tikhonov regularization to solve the inverse problems and study its convergence analysis in the Banach space setting, where the iterates of proposed method are defined through minimization problems. These minimization problems involve uniformly convex penalty terms that may be non-smooth, which can be utilized in the reconstruction of several notable features (e.g., discontinuities and sparsity) of the sought solutions. We terminate our method through a discrepancy principle and show its regularizing nature. Moreover, we show the convergence of the iterates of the method with respect to Bregman distance as well as their strong convergence. In order to show the applicability of the proposed method, we show that it is applicable on two ill-posed inverse problems. Finally, by assuming that a stability estimate holds within a certain set, we derive the convergence rates of our method.

On learning rates for regularized Nyström subsampling in unsupervised domain adaptation

The regularized Nyström subsampling is a popular approach for learning problems that deals with big data. We employ such technique in the context of the unsupervised domain adaptation problems with covariate shift assumption. Within the framework of the Reproducing Kernel Hilbert Space concept, an algorithm is constructed that is a combination of the Nyström subsampling and the two-steps iterated Tikhonov regularization. This approach allows significantly reduce the amount of computing resources involved and at the same time maintains the same learning rates as for the standard machine learning algorithms.

Electrical capacitance tomography image reconstruction based on iterative Tikhonov regularization improved algorithm

Electrical capacitance tomography image reconstruction based on iterative Tikhonov regularization improved algorithm

Simplified REGINN-IT method in Banach spaces for nonlinear ill-posed operator equations

Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.

A relaxed iterated Tikhonov regularization for linear ill-posed inverse problems

Iterated Tikhonov regularization methods are prevalent regularization methods that can make the solution of ill-posed problems less sensitive to noise. As the iteration number grows, their solution gradually converges to the optimum approximation of the desired solution but then changes to the worse, causing a semiconvergence. This difficulty can be reduced by a reliable early termination rule or a strategy for choosing the appropriate regularization parameter at each iteration. Different from the existing methods, this paper describes a novel iterative Tikhonov regularization based on introducing a non-decreasing sequence of relaxation parameters into the stationary iterated Tikhonov regularization, providing a fast and roughly stable convergence. After investigating theoretically the convergence rate of the novel iterative Tikhonov regularization, we propose a series of numerical experiments for the evaluation of its accuracy and finally describe a real-world application in the field of 3-dimensional image restoration problems to verify its efficiency.

Assessment of the double-parameter iterative Tikhonov regularization for single-epoch measurement model-based precise GNSS positioning

The double-parameter iterative Tikhonov regularization is assessed for strengthening the single-epoch model-based precise GNSS positioning within the framework of cognitive meanings. The simultaneous iterative Tikhonov regularization of the least-squares (LS) estimators of the parameters of interest in the single-epoch GNSS model is analyzed to enhance their accuracy properties. Regularization parameters (RP) are collected in the regularization operator, which can play a standardizing role in the regularization principle. Thus, the double-parameter approach can consider the heteroscedasticity of the LS estimators of interest in the regularization principle. This approach employs the quality-based mean squared error (mse) matrix trace minimization criterion to select optimal double-RP values simultaneously. Then, the unconstrained LS estimation is iteratively regularized, obtaining regularized float solutions. The numerical tests indicate that the double-parameter approach successfully strengthens the single-epoch GNSS measurement models due to considering the heteroscedasticity of the LS estimators of interest in the regularization principle. The regularized variance–covariance (vc) matrix describes float solutions of improved precision properties at the cost of losing the LS estimator’s unbiasedness. The accuracy is thus superior in the sense of mse. Therefore, the regularized LS estimator is well-scaled but with a biased localization. It also provides a more peaked probability density function (PDF) of real-valued ambiguities, obtaining a lower failure rate (FR) of integer least-squares (ILS) ambiguity resolution (AR). In this way, the regularized ILS estimator performs well in the ambiguity domain with the presence of regularized bias.

A COLLOCATION METHOD FOR FREDHOLM INTEGRAL EQUATIONS OF THE FIRST KIND VIA ITERATIVE REGULARIZATION SCHEME

To solve the ill-posed integral equations, we use the regularized collocation method. This numerical method is a combination of the Legendre polynomials with non-stationary iterated Tikhonov regularization with fixed parameter. A theoretical justification of the proposed method under the required assumptions is detailed. Finally, numerical experiments demonstrate the efficiency of this method.

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.

A BIM-Based Algorithm for Quantitative Monitoring of Temperature Distribution During Breast Hyperthermia Treatments

Microwave hyperthermia (MH) treatment for breast cancer is a research interest due to its capability to initiate cell necrosis in malignant tumor or to enhance the effect of other treatment modalities such as chemotherapy. The goal of MH treatment is to increase temperature of malignant tumor up to 45°C based on the treatment plan; however, microwave energy focusing is a challenging problem and may cause unwanted hotspots on healthy tissues; therefore, there is a need to monitor the temperature. In this paper, an iterative differential microwave imaging algorithm for temperature monitoring is presented. The algorithm is based on Born iterative method (BIM) and Tikhonov regularization. Feasibility of the algorithm is shown by a large computational study using realistic digital breast phantoms via TM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> polarized 2-D scattered fields. Also, some results are given for calibrated scattering parameters, which are obtained from both a 3-D electromagnetic simulation program and a simple measurement setup. An approach for selection of matching medium in hyperthermia monitoring applications is also presented. The reconstructions are performed with scattered field data collected at 11 discrete frequency points uniformly taken from the 0.5-1.5 GHz range. For a specific heating scenario in 2-D problem, reconstruction error is lower than 0.3% with ±10% noise on reference dielectric property distribution and 40 dB signal to noise ratio (SNR). The results show that the proposed approach provides up to 3°C and 0.1°C resolution in temperature estimation with ±10% noise on reference dielectric property distribution for 30 dB and 60 dB SNR values, respectively.

An efficient one-step proximal method for EIT sparse reconstruction based on nonstationary iterated Tikhonov regularization

ABSTRACT Image reconstruction of EIT mathematically is a seriously ill-posed inverse problem, which is easily affected by measurement noise. This fact leads to a relatively low spatial resolution, particularly in the accuracy of identifying object boundaries. This paper concerns with reconstructing a finite number of simple and small inclusions equipped with the homogeneous background conductivity. The sparsity with respect to the inclusions in the spacial representation domain is apriori assumed. We try to reexamine the multi-parameter constraints on mixing between -norm and -norm for attaining the sparse reconstruction as well as simultaneously enhancing the stability. On the basis of the reference approximation reconstructed by nonstationary iterated Tikhonov regularization (NITR) method, we propose a novel one-step proximal sparsity-promoting approach by one induced proximal shrink operator, abbreviated as one-step PNITR method. The proposed one-step PNITR method consists of twofold: one first performs NITR with mth iteration to generate the reference approximation, and then performs one-step proximal shrinkage processing and one forcing constraint function on it to obtain the final sparsity-promoting reconstruction. For the latter, the former aims to not only enhance higher reliability but also guarantee the sparsity. The proposed method can benefit from the double regularization effect. It is supposed to further improve the imaging quality in terms of sharpening the edges and reducing the artefacts. To validate the advantage of the proposed method, numerical simulations with synthetic data have been carried out. Also, qualitative and quantitative comparisons are conducted. Results indicate that the proposed approach can lead to substantial improvements in EIT image resolution. The performed numerical simulations show that our method is promising.

Investigating the effect of oblique image acquisition on the accuracy of QSM and a robust tilt correction method.

Quantitative susceptibility mapping (QSM) is used increasingly for clinical research where oblique image acquisition is commonplace, but its effects on QSM accuracy are not well understood. The QSM processing pipeline involves defining the unit magnetic dipole kernel, which requires knowledge of the direction of the main magnetic field with respect to the acquired image volume axes. The direction of is dependent on the axis and angle of rotation in oblique acquisition. Using both a numerical brain phantom and in vivo acquisitions in 5 healthy volunteers, we analyzed the effects of oblique acquisition on magnetic susceptibility maps. We compared three tilt-correction schemes at each step in the QSM pipeline: phase unwrapping, background field removal and susceptibility calculation, using the RMS error and QSM-tuned structural similarity index. Rotation of wrapped phase images gave severe artifacts. Background field removal with projection onto dipole fields gave the most accurate susceptibilities when the field map was first rotated into alignment with . Laplacian boundary value and variable-kernel sophisticated harmonic artifact reduction for phase data background field removal methods gave accurate results without tilt correction. For susceptibility calculation, thresholded k-space division, iterative Tikhonov regularization, and weighted linear total variation regularization, all performed most accurately when local field maps were rotated into alignment with before susceptibility calculation. For accurate QSM, oblique acquisition must be taken into account. Rotation of images into alignment with should be carried out after phase unwrapping and before background-field removal. We provide open-source tilt-correction code to incorporate easily into existing pipelines: https://github.com/o-snow/QSM_TiltCorrection.git.

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.

Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stability

In this paper, we study the nonstationary iterated Tikhonov regularization method (NITRM) proposed by Jin and Zhong (2014 Numer. Math. 127 485–513) to solve the inverse problems, where the inverse mapping fulfills a Hölder stability estimate. The iterates of NITRM are defined through certain minimization problems in the settings of Banach spaces. In order to study the various important characteristics of the sought solution, we consider the non-smooth uniformly convex penalty terms in the minimization problems. In the case of noisy data, we terminate the method via a discrepancy principle and show the strong convergence of the iterates as well as the convergence with respect to the Bregman distance. For noise free data, we show the convergence of the iterates to the sought solution. Additionally, we derive the convergence rates of NITRM method for both the noisy and noise free data that are missing from the literature. In order to derive the convergence rates, we solely utilize the Hölder stability of the inverse mapping that opposes the standard analysis which requires a source condition as well as a nonlinearity estimate to be satisfied by the inverse mapping. Finally, we discuss three numerical examples to show the validity of our results.

An Arnoldi-based preconditioner for iterated Tikhonov regularization

Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel “approximate Tikhonov regularization method” based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method.

Optimized Dynamic Motion Performance for a 5-DoF Electromagnetic Manipulation

High precision manipulation and optimization of actuation current are major concerns in the design of magnetic manipulation systems. In this paper, a flexible magnetic field mapping model has been employed to determine the field distribution of a compact electromagnetic manipulation system in 3D space, and validated by a high-resolution magnetic calibration system and Finite Element Analysis, which enables an improved accuracy and performance of magnetic manipulation. In addition, an iterative current optimization scheme is used to optimize the actuation current through Iterated Tikhonov Regularization. Finally, a feedback control algorithm including a Proportional-Integral-Derivative controller and Extended State Observer is conducted to solve general external disturbance during the autonomous manipulation of magnetic objects. By incorporating magnetic field mapping, feedback control, and current optimization model, the accuracy of the system has been demonstrated by navigating a small-size permanent magnet through predefined trajectories (Z-shaped, spiral curve) with the average position error of 0.27 mm (body length of 1 mm) and maximum average velocity of 3.59 mm/s. Meanwhile, the results showed that the required maximum actuation current was reduced from 4.00 A to 1.81 A. Our work provides a reliable and systematic approach for high-precision magnetic manipulation and optimization of actuation currents.