Variational models are one of the most efficient techniques for image de-noising problems. A variational method refers to the technique of optimizing a functional in order to restore appropriate solutions from observed data that best  t the original image. This paper proposes to revisit the discrete total generalized variation (TGV ) image denoising problem by rede ning the op-erations via the inclusion of a diagonal term to reduce the staircasing effect, which is the patchy artifacts usually observed in slanted regions of the image. We propose to add an oblique scheme in discretization operators, which we claim is aware of the alleviation of the staircasing effect superior to the con-ventional TGV method. Numerical experiments are carried out by using the primal-dual algorithm, and numerous real-world examples are conducted to con rm that the new proposed method achieves higher quality in terms of rel-ative error and the peak signal to noise ratio compared with the conventional TGV method.

To address the inherent ill-posedness of the geophysical inverse problems, it is necessary to include a suitable regularization function in the corresponding optimization framework. Typically, the choice of the regularization function depends on prior assumptions about the geometric characteristics of the unknown model parameters, e. g., smoothness or blockiness. First-order total variation regularization (TV) allows the reconstruction of well-defined edges and models exhibiting block-like characteristics. However, it is associated with the generation of undesirable staircase artifacts. This study applies a novel approach for removing staircase artifacts using a combined second-order non-convex total variation with overlapping group sparse regularizer. This regularizer aims to smooth out the staircase effect while still keeping the edges of the model. Moreover, the study applies the proposed method for the nonlinear seismic cross-hole tomography problems, where the goal is to reconstruct both smooth and blocky features of the model and avoid staircase artifacts of the TV regularization. The numerical examples indicate the efficiency of the proposed regularization method.

A new hybrid variational model is presented for image denoising, which in-corporates the merits of Shannon interpolation, total generalized variation (TGV) regularization, and a symmetrized derivative regularization term based on l1-norm. In this model, the regularization term is a combination of a TGV functional and the symmetrized derivative regularization term, while the data fidelity term is characterized by the l2-norm. Unlike most variational models that are discretized using a finite-difference scheme, our approach in structure is based on Shannon interpolation. Quantitative and qualitative assessments of the new model indicate its effectiveness in restoration accuracy and staircase effect suppression. Numerical experi-ments are carried out using the primal-dual algorithm. Numerous real- world examples are conducted to confirm that the newly proposed method outperforms several current state-of-the-art numerical methods in terms of the peak signal to noise ratio and the structural similarity (SSIM) index.

The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation in order to identify its most suitable regularization method. In the geoid computations without applying Stokes formula, the downward continuation based on Abel-Poisson integral is an inverse problem, which requires regularization. Since so far the regularization of this ill-posed problem has been thoroughly studied, in this paper the regularization of the downward continuation problem based on Abel-Poisson integral, is investigated and various techniques falling into the aforementioned classes of regularizations are applied and their efficiency is compared. From the first class Truncated Singular Value Decomposition (TSVD) and Truncated Generalized Singular Value Decomposition (TGSVD) methods and from the second class Generalized Tikhonov (GT) with the norms and semi-norms in Sobolev subspaces W12(a,b), W22(a,b) are applied and their capabilities for the regularization of the problem is compared. Our numerical results derived from simulated studies reveal that the GT method with discretized norm of Sobolev subspace W22(a,b) gives the best results among the studied methods for the regularization of the downward continuation problem based on the Abel-Poisson integral. On the contrary, the TGSVD method with the discretized second order derivatives has less consistency with the ill-posed problem and yields less accuracy. Finally, the GT method with discretized norm of Sobolev subspace W22(a,b) is applied to the downward continuation of real gravity data of the type modulus of gravity acceleration within the geographical region of Iran to derive a geoid model for this region.

In this paper,  a generalized version of the auto-convolution Volterra integral equation of the first kind as an ill-posed problem is studied. We apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. According to the proposed numerical method, for $n=0$ and  $n=1,\ldots, N-1$, we obtain a  nonlinear and linear system, respectively. We have to distinguish between two cases, nonlinear and linear systems of algebraic equations. A double iteration process based on the modified Tikhonov regularization method is considered to solve the nonlinear algebraic equations. In this process, the outer iteration controls the evolution path of the unknown vector $U_0^{\delta}$ in the selected direction $\tilde{u}_0$, which is determined from the inner iteration process. For the linear case, we apply the Lavrentiev $\tilde{m}$ times iterated regularization method to deal with the ill-posed linear system. The validity and efficiency of the proposed method are demonstrated by several numerical experiments.

When the molecules and atoms of the atmosphere receive enough external energy, one or more electrons are dissociated from the molecules or atoms. This process is called ionization. The solar ultraviolet (EUV) radiation and particle precipitation are the two primary energy sources in the ionization. Also cosmic radiation contributes to this ionization. This layer of atmosphere is called ionosphere. The ionosphere is that part of the atmosphere in which the number of free electrons is so high that, it significantly affects the propagation of radio waves. Ionospheric refraction is one of the main error sources on GPS signals. This effect is proportional to the total electron content (TEC). TEC is a projection of electron density along signal path extending from the satellite to the receiver on the ground. The unit of TEC is TECU and 1 TECU equals 1016 electrons/m2. Production of free electrons in the ionosphere depends on many factors, such as solar, geomagnetic, gravitational and seismic activity period.There are many methods to obtain electron density or TEC profiles and predictions. In early time, direct measurements such as ionosonde was a kind of effective instrument to achieve this purpose. Later, some empirical and mathematical models were developed. For example, IRI (international reference ionosphere) model, PIM (the parameterized ionospheric model) are empirical models.Mathematical models divided to two categories: single-layer (2-D) and multi-layer (3-D & 4-D). The existing 2-D methods of modeling the electron density can be classified to non-grid based and grid based techniques. The former modeling techniques are based on the least squares estimation of a functional model for certain types of observables derived from the GPS carrier phase and code measurements. Polynomials and spherical harmonics are some of the base functions that are commonly in use. In grid based modeling, the spherical shell of free electrons is developed into a grid of rectangular elements. Special reconstruction algorithms are then used for estimating the electron density within the every element of the shell.Neglecting the vertical gradient of the electron density is the main deficiency of the two dimensional modeling techniques. To study the physical properties of the ionosphere, computerized tomography (CT) demonstrated an efficient and effective manner. Due to the sparse distribution of GPS stations and viewing angle limitations, ionospheric electron density (IED) reconstruction is an ill-posed inverse problem. Usually, iterative or non- iterative algorithm used for electron density reconstruction. Non- iterative algorithms are known regularization methods.Using these methods to solve the ill posed problems will produce bias in final results. In this paper, we used hybrid regularization algorithm for solving ionosphere tomography. This method is a combination of two regularizations methods: Tikhonov regularization and total variation (TV). Tikhonov regularization is a classical method for solving ill-posed inverse problem and total variation effectively resists noise in results. To apply the method for constructing a 3D-image of the electron density, GPS measurements of the Iranian permanent GPS network (at 3-day in 2007) have been used. The modeling region is between 240 to 400 N and 440 to 640 W. The result of hybrid regularization method has been compared to that of the zero order Tikhonov regularization method and NeQuick model outputs. The minimum relative error for hybrid method is 1.55% and the maximum relative error is 19.52%. Also, maximum and minimum absolute error is computed 1.32´1011 (ele/m3) and 6.67´1011 (ele/m3), respectively.Experiments demonstrate the effectiveness, and illustrate the validity and reliability of the proposed method.