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Satellite microwave instruments have different field of views (FOVs) in different channels. A direct average technique (“direct method”) is frequently used to generate gridded datasets in the earth science community. A large FOV will measure radiance from outside the area of a designated grid cell. Thus, the direct method will lead to errors in a measurement over a grid cell because some pixels covering areas outside of the cell are involved in the averaging process. The Backus-Gilbert method (BG method) is proposed and demonstrated to minimize those uncertainties. Three sampling resolutions (6.5 km × 6.0 km, 11.5 km × 6.0 km, 13.0 km × 6.0 km) are analyzed based on the scanning characteristics of the Global Precipitation Measurement (GPM) Microwave Imager (GMI) 18.9-GHz channel. Brightness temperatures (TBs) at 0.5 km × 0.5 km resolution over eastern China are used to obtain synthetic 18.9-GHz TBs at the three sampling resolutions. The direct and BG methods are both applied to create a 25 km × 25 km gridded dataset and their related uncertainties are analyzed. Results indicate the error variances with the direct method are 3.00, 3.68 and 4.99 K2 at the three sampling resolutions, respectively. By contrast, the BG method leads to a much smaller error variance than the direct method, especially over areas with a large TB gradient. Two GMI orbital measurements are applied to verify the BG method for gridding process is reliable. The BG method could be utilized for general purpose of creating a gridded dataset.

形態: カラー図版あり, Physical characteristics: Original contains color illustrations, Accepted: 2019-11-21, 資料番号: PA2010013000

In this paper, we study distributed learning with multi-penalty regularization based on a divide-and-conquer approach. Using Neumann expansion and a second order decomposition on difference of operator inverses approach, we derive optimal learning rates for distributed multi-penalty regularization in expectation. As a byproduct, we also deduce optimal learning rates for multi-penalty regularization, which was not given in the literature. These results are applied to the distributed manifold regularization and optimal learning rates are given.

We apply the Backus-Gilbert approach to normal mode center frequency data, to constrain jumps in P, S, bulk-sound speed and density at the "660" discontinuity in the earth's mantle (~650-670 km dep...

We investigate the use of a functional analytical version of the Backus-Gilbert Method as a reconstruction strategy to get specific information about the solution of linear and slightly non-linear systems with Frech\'et derivable operators. Some a priori error estimates are shown and tested for two classes of problems: a nonlinear moment problem and a linear elliptic Cauchy problem. For this second class of problems a special version of the Green-formula is developed in order to analyze the involved adjoint equations., Comment: 15 pages, 6 figures

We consider regularization methods based on the coupling of Tikhonov regularization and projection strategies. From the resulting constraint regularization method we obtain level set methods in a straight forward way. Moreover, we show that this approach links the areas of asymptotic regularization to inverse problems theory, scale-space theory to computer vision, level set methods, and shape optimization., Comment: 12 pages, Letter to the editor

International audience; Dynamic forces reconstruction from vibration data is an ill-posed inverse problem. A standard approach to stabilize the reconstruction consists in using some prior information on the quantities to identify. This is generally done by including in the formulation of the inverse problem a regularization term as an additive or a multiplicative constraint. In the present article, a space-frequency multiplicative regularization is developed to identify mechanical forces acting on a structure. The proposed regularization strategy takes advantage of one's prior knowledge of the nature and the location of excitation sources, as well as that of their spectral contents. Furthermore, it has the merit to be free from the preliminary definition of any regular-ization parameter. The validity of the proposed regularization procedure is assessed numerically and experimentally. It is more particularly pointed out that properly exploiting the space-frequency characteristics of the excitation field to identify can improve the quality of the force reconstruction.

In this paper, we consider a time-fractional inverse diffusion problem, where the data are given at x = 1 and the solution is required in the interval 0 x 1 . This problem is typically ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The simplified Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, a new a posteriori parameter choice rule is proposed and the Holder type error estimate is also obtained. Some different type examples are presented to demonstrate the feasibility and efficiency of the proposed method.

Computing steering control for an approximately controllable linear system for a given target state is an ill-posed problem. We use a weighted Tikhonov regularization method and compute the regularized control. It is proved that the target state corresponding to the regularized control is close to the actual state to be attained. We also obtained error estimates and convergence rates involved in the regularization procedure using both the a priori and a posteriori parameter choice rule. Theory is substantiated with numerical experiments.

The worst-case behaviour of a general class of regularization algorithms is considered in the case where only objective function values and associated gradient vectors are evaluated. Upper bounds are derived on the number of such evaluations that are needed for the algorithm to produce an approximate first-order critical point whose accuracy is within a user-defined threshold. The analysis covers the entire range of meaningful powers in the regularization term as well as in the Hölder exponent for the gradient. The resulting complexity bounds vary according to the regularization power and the assumed Hölder exponent, recovering known results when available.

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix and an error-contaminated data vector (right-hand side). This regularization method replaces the given problem by a penalized least-squares problem. It is well known that Tikhonov regularization in standard form may yield approximate solutions that are too smooth, i.e., the computed approximate solution may lack many details that the desired solution of the associated, but unavailable, error-free problem might possess. Fractional Tikhonov regularization methods have been introduced to remedy this shortcoming. However, the computed solution determined by fractional Tikhonov methods in standard form may display undesirable spurious oscillations. This paper proposes that fractional Tikhonov methods be equipped with a nonlinear penalty term, such as a TV-norm penalty term, to reduce unwanted oscillations. Numerical examples illustrate the benefits of this approach.

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analys...

We consider the solution of large linear systems of equations that arise from the discretization of ill-posed problems. The matrix has a Kronecker product structure and the right-hand side is contaminated by measurement error. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel and in image restoration problems. Regularization methods, such as Tikhonov regularization, have to be employed to reduce the propagation of the error in the right-hand side into the computed solution. We investigate the use of the global GolubKahan bidiagonalization method to reduce the given large problem to a small one. The small problem is solved by employing Tikhonov regularization. A regularization parameter determines the amount of regularization. The connection between global GolubKahan bidiagonalization and Gauss-type quadrature rules is exploited to inexpensively compute bounds that are useful for determining the regularization parameter by the discrepancy principle.

Computation of control for a controlled partial differential equation is a difficult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regula...

In this study Tikhonov Gradient type-method is considered for nonlinear ill-posed operator equations. In our convergence analysis, we use hypotheses only on the first Frechet derivative of F in contrast to the higher order Frechet derivatives used in the earlier studies. We obtained ‘optimal’ order error estimate by choosing the regularization parameter according to the adaptive method proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005).

I have developed an improved practical method for interpreting a symmetrical-shaped potential field anomaly due to an isolated source body of regular geometric configuration. The method uses the first-order horizontal derivative of the logarithmically transformed absolute value of the anomaly in estimating the source-body parameters, such as the location, depth of burial, and shape factor. To tackle noise in data, a regularization technique is designed, which ensures a robust estimate of the first-order derivative of logarithmically transformed data. The regularization technique uses an optimal value of regularization parameter that, although noise dependent, requires no a priori knowledge of the noise level in the data. A graphical method is designed to determine an optimal value of the regularization parameter from the position of the local minimum of a specially defined functional with respect to the regularization parameters. Numerical tests have been conducted on the noise-contaminated synthetic data to validate the proposed method. The successful application of the method on published field data for the gravity and magnetic anomaly suggests the applicability of the method.

In this study, we apply Tikhonov’s regularization algorithm for a 3-D density inversion from the gravity-gradiometry data. To reduce the non-uniqueness of the inverse solution (carried out without additional information from geological evidence), we implement the depth-weighting empirical function. However, the application of an empirical function in the inversion equation brings the bias problem of the regularization factor when a traditional Tikhonov’s algorithm is applied. To solve the bias problem of regularization factor selection, we present a standardized solution that comprises two parts for solving a 3-D constrained inversion equation, specifically the linear matrix transformation and Tikhonov’s regularization algorithm. Since traditional regularization techniques become numerically inefficient when dealing with large number of data, we further apply methods which include the Simultaneous Iterative Reconstruction Technique (SIRT) and the wavelet compression combined with Least Squares QR-decomposition (LSQR). In our simulation study, we demonstrate that SIRT as well as the wavelet compression plus LSQR algorithm improve the computation efficiency, while provide results which closely agree with that obtained from applying Tikhonov’s regularization. In particular, the algorithm of wavelet compression plus LSQR shows the best computing efficiency, because it combines the advantages of coefficients compression of big matrix and fast solution of sparse matrix. Similar findings are confirmed from the vertical gravity gradient data inversion for detecting potential deposits at the Kauring (near Perth, Western Australia) testing site.

Ill-posed problems are widely existed in signal processing. In this paper, we review popular regularization models such as truncated singular value decomposition regularization, iterative regularization, variational regularization. Meanwhile, we also retrospect popular optimization approaches and regularization parameter choice methods. In fact, the regularization problem is inherently a multi-objective problem. The traditional methods usually combine the fidelity term and the regularization term into a single-objective with regularization parameters, which are difficult to tune. Therefore, we propose a multi-objective framework for ill-posed problems, which can handle complex features of problem such as non-convexity, discontinuity. In this framework, the fidelity term and regularization term are optimized simultaneously to gain more insights into the ill-posed problems. A case study on signal recovery shows the effectiveness of the multi-objective framework for ill-posed problems.

We present a fast computational framework for the inverse medium problem in scattering, i.e. we look at discretization, reconstruction and numerical performance. The Helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements. For the reconstruction of the medium, we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters. The underlying paradigm is, roughly speaking, to minimize the discrepancy between the reconstruction and measured data while, at the same time, taking into account various structural a-priori information via suitable penalty terms. In particular, the involved penalty terms are designed to promote information expected in real-world environments. To this end, a combination of sparsity promoting terms, total variation, and physical bounds of the inhomogeneous medium, e.g. positivity constraints, is employed in the regularization penalty. A primal-dual algorithm is used to solve the minimization problem related to the variational regularization. The computational feasibility, performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data.

Tikhonov regularization is commonly used for the solution of linear discrete ill-posed problems with error-contaminated data. A regularization parameter that determines the quality of the computed solution has to be chosen. One of the most popular approaches to choosing this parameter is to minimize the Generalized Cross Validation (GCV) function. The minimum can be determined quite inexpensively when the matrix A that defines the linear discrete ill-posed problem is small enough to rapidly compute its singular value decomposition (SVD). We are interested in the solution of linear discrete ill-posed problems with a matrix A that is too large to make the computation of its complete SVD feasible, and show how upper and lower bounds for the numerator and denominator of the GCV function can be determined fairly inexpensively for large matrices A by computing only a few of the largest singular values and associated singular vectors of A. These bounds are used to determine a suitable value of the regularization parameter. Computed examples illustrate the performance of the proposed method.

Inverse problems of identifying parameters in partial differential equations constitute an important class of problems with diverse real-world applications. These identification problems are commonly explored in an optimization framework and there are many optimization formulations having their own advantages and disadvantages. Although a non-convex output least-squares (OLS) objective is commonly used, a convex-modified output least-squares (MOLS) has shown encouraging results in recent years. In this work, we focus on various aspects of the MOLS approach. We devise a rigorous (quadratic and non-quadratic) regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying sharply varying coefficients and also in image processing. We give new existence results for the regularized optimization problems for OLS and MOLS. Restricting to the Tikhonov (qua...

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence $$O(\delta ^\frac{\alpha +1}{\alpha +2})$$ for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results.

źp regularization has been a popular pruning method for neural networks. The parameter p was usually set as 0 < p ź 2 in the literature, and practical training algorithms with ź0 regularization are lacking due to the NP-hard nature of the ź0 regularization problem; however, the ź0 regularization tends to produce the sparsest solution, corresponding to the most parsimonious network structure which is desirable in view of the generalization ability. To this end, this paper considers an online gradient training algorithm with smoothing ź0 regularization (OGTSL0) for feedforward neural networks, where the ź0 regularizer is approximated by a series of smoothing functions. The underlying principle for the sparsity of OGTSL0 is provided, and the convergence of the algorithm is also theoretically analyzed. Simulation examples support the theoretical analysis and illustrate the superiority of the proposed algorithm.

In this paper, a numerical method is proposed for the numerical solution of a linear system with suitable initial and boundary conditions using the cubic B-spline collocation scheme to determine th...

Numerical reconstruction of the missing boundary data in 2D and 3D linear elasticity.Noisy over-prescribed data are available on the remaining accessible boundary.The inverse problem is solved by a meshless method, i.e. the method of fundamental solutions (MFS).Stabilization is achieved via several SVD-based regularization methods.The regularization parameter is selected via the criteria used for inverse problems. The numerical reconstruction of the missing Dirichlet and Neumann data on an inaccessible part of the boundary in the case of two- and three-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part is investigated. This inverse problem is solved using the method of fundamental solutions (MFS), whilst its stabilization is achieved through several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization methodź48, the damped SVD and the truncated SVDź18. The regularization parameter is selected according to the discrepancy principleź40, generalized cross-validation criterionź14 and Hansen's L-curve methodź20.

We treat the estimation of a sparse set of sources emitting plane waves observed by a sensor array as a complex-valued LASSO (c-LASSO) problem where the usual ź1-norm constraint is replaced by the ź1-norm of a matrix D times the solution vector. When the sparsity order is given, algorithmically selecting a suitable value for the c-LASSO regularization parameter remains a challenging task. The corresponding dual problem is formulated and it is shown that the dual solution is useful for selecting the regularization parameter of the c-LASSO. The solution path of the c-LASSO is analyzed and this motivates an order-recursive algorithm for the selection of the regularization parameter and a faster iterative algorithm that is based on a further approximation. This greatly facilitates computation of the c-LASSO-path as we can predict the changes in the active indices as the regularization parameter is reduced. Using this regularization parameter, the directions of arrival for all sources are estimated. HighlightsThe dual problem to the complex-valued c-LASSO is formulated and analyzed.The dual solution is the output of a weighted matched filter of c-LASSO residuals.The dual solution motivates a selection of the c-LASSO regularization parameter.This gives an order-recursive scheme for selecting the regularization parameter.Approximations lead to the formulation of faster iterative algorithms for its solution.

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, based on the derived expressions, we prove structured condition numbers are smaller than their corresponding unstructured counterparts. By means of the power method and small sample statistical condition estimation (SCE), fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). For large scale linear structured Tikhonov regularization problems, we show how to incorporate the SCE into the preconditioned conjugate gradient (PCG) method to get the posterior error estimations. The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable. Also, an image restoration example is tested to show the effectiveness of the SCE for large scale linear structured Tikhonov regularization problems.

First-arrival traveltime tomography is a robust tool for near-surface velocity estimation. A common approach to stabilizing the ill-posed inverse problem is to apply Tikhonov regularization to the inversion. However, the Tikhonov regularization method recovers smooth local structures while blurring the sharp features in the model solution. We present a first-arrival traveltime tomography method with modified total-variation regularization to preserve sharp velocity contrasts and improve the accuracy of velocity inversion. To solve the minimization problem of the new traveltime tomography method, we decouple the original optimization problem into the two following subproblems: a standard traveltime tomography problem with the traditional Tikhonov regularization and a L2 total-variation problem. We apply the conjugate gradient method and split-Bregman iterative method to solve these two subproblems, respectively. Our synthetic examples show that the new method produces higher resolution models than the conventional traveltime tomography with Tikhonov regularization, and creates less artefacts than the total variation regularization method for the models with sharp interfaces. For the field data, pre-stack time migration sections show that the modified total-variation traveltime tomography produces a near-surface velocity model, which makes statics corrections more accurate.

In this paper, we study a fractional Tikhonov regularization method (FTRM) for solving a Cauchy problem of Helmholtz equation in the frequency domain. On the one hand, the FTRM retains the advantage of classical Tikhonov method. On the other hand, our method can prevent the effect of oversmoothing of classical Tikhonov method and conveniently control the amount of damping. The convergence error estimates between the exact solution and its regularization approximation are constructed. Several interesting numerical examples are provided, which validate the effectiveness of the proposed method.

In this paper, we study a non-characteristic Cauchy problem for a radially symmetric inverse heat conduction equation in a two-layer domain. This is a severely ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. It is well-known that the classical Tikhonov regularization solutions are too smooth and the approximate solutions may lack details that might be contained in the exact solutions. Combining Fourier transform technique with a modified version of the classical Tikhonov regularization, we obtain a regularized solution which is stably convergent to the exact solution with a sharp error estimate.

In this paper we establish the error estimates for multi-penalty regularization under the general smoothness assumption in the context of learning theory. One of the motivation for this work is to study the convergence analysis of two-parameter regularization theoretically in the manifold learning setting. In this spirit, we obtain the error bounds for the manifold learning problem using more general framework of multi-penalty regularization. We propose a new parameter choice rule "the balanced-discrepancy principle" and analyze the convergence of the scheme with the help of estimated error bounds. We show that multi-penalty regularization with the proposed parameter choice exhibits the convergence rates similar to single-penalty regularization. Finally on a series of test samples we demonstrate the superiority of multi-parameter regularization over single-penalty regularization.

In this paper we use the L-curve method and the Morozov discrepancy principle for the estimation of the regularization parameter in the regularization of time-delayed optimal control computation. Zeroth order, first order and second order differential operators are considered. Two test examples show that the L-curve method and the two discrepancy principles give close estimations for the regularization parameters.

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify ...

Electrical resistance tomography (ERT) reconstructs the conductivity distribution from the boundary changes of electrical measurements. The inverse problem of ERT is seriously ill-posed where regularization methods are needed to treat this ill-posedness. A proper choice of regularization parameter which controls the degree of smoothing is very important for these regularization methods. Although have been a variety of methods, such as L-curve method, to choose a reasonable parameter for the problem, these methods usually result in a scalar parameter which cannot distinctly express the spatial characteristic of the conductivity distribution. So a spatially adaptive regularization parameter choice method is proposed for regularizing the inverse problem of ERT based on Tikhonov regularization. Since large regularization parameters can stabilize and smoothen the solution, while small regularization parameters can approximate and sharpen the solution, the proposed method adaptively updates the regularization parameters during the iteration process and provides spatially varying parameter for each pixel of the reconstructed image. When the iteration is stopped, large regularization parameters for the smooth background region and small regularization parameters for the object region can be obtained. The method is discussed using simulated data for some typical conductivity distributions, and further applied to the analysis of real measurement data acquiring from the practical system. The results demonstrate that flexible regularization parameter vectors can be achieved for different distributions and the strength of regularization is adaptively provided for different regions in a specific distribution. The adaptive method achieves an efficient and reliable regularization solution and has outstanding performance in noise immunity especially in smooth background regions.

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods.

One promising technique for single image super-resolution (SR) is reconstruction-based framework, where the key issue is to apply reasonable prior knowledge to well pose the solution to upsampled images. In this paper, we employ the non-local steering kernel regression (NLSKR) model to devise an effective regularization term for solving single image SR problem. The proposed regularization term is based on the complementary properties of local structural regularity and non-local self-similarity existing in natural images, aiming at preserving sharp edges and producing fine details in the resultant image. By integrating the regularization term into the standard back-projection framework, we solve a least squares minimization problem to seek the desired high-resolution (HR) image. Extensive experimental results on several public databases indicate that the proposed method produces promising results in terms of both objective and subjective quality assessments. HighlightsThe non-local steering kernel regression (NLSKR) model is used to devise an effective regularization term for solving the single image SR problem.A new regularization term is designed for preserving sharp edges and producing fine details in the resultant image.By integrating the regularization term into the standard back-projection framework, we solve a least squares minimization problem to seek the desired high-resolution image.

Descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems. They are either directly applied to the discrepancy term, which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms. In its basic form, gradient descent methods converge slowly.We aim at extending different optimization schemes, which have been recently introduced for accelerating these approaches, by addressing more general penalty terms. In particular we use a general setting in infinite Hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints.To illustrate the efficiency of these algorithms, we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization.

Superior to the other well-known iterations, such as Landweber iteration, total variational regularization is a well-established method for identifying the piecewise smooth parameter and processing the image. Therefore, we propose a new iteratively total variational regularization based on total variational regularization and homotopy perturbation technique in this paper. Taking advantage of Bregman distance, we construct a nested iteration to inverse parameters. Meanwhile we prove that the error of the new method decreases monotonically, and the new iterative method with the noisy data is regular according to the discrepancy principle. In the last numerical section, compared with Landweber type total variational regularization and Runge-Kutta type regularization, numerical results of this new method indicate that this new regularization is time saving for the same accuracy, and more robust to the noisy data.

In this paper, we consider a Cauchy problem for the semi-linear elliptic equation and use a modified boundary Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability of the regularization solution are proven. Under an a-priori bound assumption for the exact solution, a convergence estimate of Holder type for this method is obtained. Finally, an iterative scheme is proposed to calculate the regularization solution, numerical results show that this method works well.

Inverse scattering problems suffer from ill-posedness and ill-conditioning, necessitating the use of regularization methods to get meaningful solutions. Commonly used regularizations are $L2$ norm based, but these generate over-smooth solutions. We propose a regularization method using both the $L1$ and $L2$ norms to obtain sharp object boundaries, while also achieving good interior reconstruction of the object permittivity. Knowledge about the permittivity can also be used as a priori information. The applicability of the method is demonstrated on synthetically generated data for two-dimensional (2-D) microwave imaging using the Born-iterative method (BIM). The optimization routine systematically estimates all parameters, while minimizing the cost function. Different objects chosen to represent realistic features have been considered to evaluate the performance. The reconstructed images indicate that the method can produce accurate object localization, shape identification, and good permittivity estimation.

The set of linear equations in the inversion of particle size distribution (PSD) based on forward light scattering is an ill-posed problem. In order to solve the inverse problem of this kind, a number of inversion algorithms have been proposed. The regularization algorithm can reconstruct the PSD, but in usual case, the solution may contain negative values and is strongly oscillating. Owing to the natural reason, the solution should be non-negative and smooth. In this paper, a simple non-negative constraint (NNC) is used with a combination of the Tikhonov regularization. Simulations and experiments show that the regularization with NNC can achieve more reasonable results.

In this paper we consider projection techniques to obtain the finite dimensional realization of a Tikhonov gradient type-method considered in George et al. (Local convergence of a Tikhonov gradient type-method under weak conditions, communicated, 2016) for approximating a solution $$\hat{x}$$ of the nonlinear ill-posed operator equation $$F(x)=y$$ . The main advantage of the proposed method is that the inverse of the operator F is not involved in the method. The regularization parameter is chosen according to the adaptive method considered by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005). We also derive optimal stopping conditions on the number of iterations necessary for obtaining the optimal order of convergence. Using two numerical examples we compare our results with an existing method to justify the theoretical results.

We investigate the optimum correction of an absolute value equation by minimally changing the coefficient matrix and right-hand side vector using Tikhonov regularization. Solving this problem is equivalent to minimizing the sum of fractional quadratic and quadratic functions. The primary difficulty with this problem is its nonconvexity. Nonetheless, we show that a global optimal solution to this problem can be found by solving an equation on a closed interval using the subgradient method. Some examples are provided to illustrate the efficiency and validity of the proposed method.

Purpose – The linear regression technique is widely used to determine empirical parameters of fatigue life profile while the results may not continuously depend on experimental data. Thus Tikhonov-Morozov method is utilized here to regularize the linear regression results and consequently reduces the influence of measurement noise without notably distorting the fatigue life distribution. The paper aims to discuss these issues. Design/methodology/approach – Tikhonov-Morozov regularization method would be shown to effectively reduce the influences of measurement noise without distorting the fatigue life distribution. Moreover since iterative regularization methods are known to be an attractive alternative to Tikhonov regularization, four gradient iterative methods called as simple iteration, minimum error, steepest descent and conjugate gradient methods are examined with an appropriate initial guess of regularized coefficients. Findings – It has been shown that in case of sparse fatigue life measurements, linear regression results may not have continuous dependence on experimental data and measurement error could lead to misinterpretations of the solution. Therefore from engineering safety point of view, utilizing regularization method could successfully reduce the influence of measurement noise without significantly distorting the fatigue life distribution. Originality/value – An excellent initial guess for mixed iterative-direct algorithm is introduced and it has been shown that the combination of Newton iterative approach and Morozov discrepancy principle is one of the interesting strategies for determination of regularization parameter having an excellent rate of convergence. Moreover since iterative methods are known to be an attractive alternative to Tikhonov regularization, four gradient descend methods are examined here for regularization of the linear regression problem. It has been found that all of gradient decent methods with an appropriate initial guess of regularized coefficients have an excellent convergence to Tikhonov-Morozov regularization results.

We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers., Comment: 28 pages, 4 figures

In this paper, we propose an efficient regularization technique (The Method of Regularized Ratios) for the reconstruction of the shape of a rigid elastic scatterer from far field measurements. The approach used is based on the factorization method and creates via Picard`s condition ratios, baptized Regularized Ratios, that serve to effectively remove unwanted singular values that may lead to poor reconstructions. This is achieved through the use of a sophisticated algorithm that progressively adjusts an initially set moderate tolerance. In comparison with the well established Tikhonov-Morozov regularization techniques our new algorithm appears to be more computationally efficient as it doesn`t require computation of the regularization parameter for each point in the grid.

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be found is smooth or discontinuous. Tikhonov regularization with appropriate regularizing functionals is applied to solve these problems. In particular, discontinuous Young’s modulus distribution can be found on the class of 2D functions with bounded variation of Hardy–Krause type. It is shown in the paper that a variant of Tikhonov regularization provides for such discontinuous distributions the so-called piecewise uniform convergence of approximate solutions as the error levels of the data vanish. The problem of practical a posteriori estimation of the accuracy for obtained approximate solutions is under consideration as well. A method of such estimation is presente...