In this study, a numerical approach of the spectral collocation method coupled with a Regularization technique is applied for solving an inverse parabolic problem of the heat equation in a quarter plane. The problem includes the estimation of an unknown boundary condition from an overspecified condition. The stable solution of the problem exists and is proved by Tikhonov Regularization technique. The algorithm works without any mesh points or elements, and accurate results can be obtained efficiently. By employing the numerical algorithm on the problem, the resultant matrix equation is ill-condition. To regularize this matrix equation, we apply Regularization technique, with the L-curve and general cross-validation criteria for choosing the Regularization parameter. For demonstrating the performance and ability of the proposed algorithm, a test example is presented. The numerical results showed that the solution obtained with the algorithm designed in this paper is stable with the noisy data and the unknown boundary condition was recovered very well.

This study deals with the 3D recovering of magnetic susceptibility model by incorporating the sparsity-based constraints in the inversion algorithm. For this purpose, the area under prospect was divided into a large number of rectangular prisms in a mesh with unknown susceptibilities. Tikhonov cost functions with two sparsity functions were used to recover the smooth parts as well as the sharp boundaries of model parameters. A pre-selected basis namely wavelet can recover the region of smooth behaviour of susceptibility distribution while Haar or finite-difference (FD) domains yield a solution with rough boundaries. Therefore, a regularizer function which can benefit from the advantages of both wavelets and Haar/FD operators in representation of the 3D magnetic susceptibility distributionwas chosen as a candidate for modeling magnetic anomalies. The optimum wavelet and parameter b which controls the weight of the two sparsifying operators were also considered. The algorithm assumed that there was no remanent magnetization and observed that magnetometry data represent only induced magnetization effect. The proposed approach is applied to a noise-corrupted synthetic data in order to demonstrate its suitability for 3D inversion of magnetic data. On obtaining satisfactory results, a case study pertaining to the ground based measurement of magnetic anomaly over a porphyry-Cu deposit located in Kerman providence of Iran. Now Chun deposit was presented to be 3D inverted. The low susceptibility in the constructed model coincides with the known location of copper ore mineralization.

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.

In the present article an energy distribution function of heterogeneous solid was estimated. Energy distribution function is an important characterization for heterogeneous adsorbent. An overall adsorption quantity for a heterogeneous solid is usually expressed by a first kind of Fredholm equation, which contains unknown distribution function and local adsorption isotherm as a kernel. The calculation of this distribution function is an ill-posed problem. The current article shows that the difficulties arising from the ill-posed nature of an adsorption equation can be overcome with the linear Regularization method and inverse theory. Performance of the Regularization method for calculation multipeak energy distribution functions' was examined in the present work. The results expressed with different charts and several random errors. The results showed that linear Regularization method is very convenient for prediction of energy distribution function of heterogeneous solids. Furthermore, if a large amount of data at low pressure is available, the performance of this method would be very suitable.Therefore, in the most cases that some data have a large random error (about 50%), Regularization method can predict the energy distribution function satisfactorily.

In this article, the application of discrete molli, cation as a Regularization procedure for solving a nonlinear inverse problem in one dimensional space is considered. Ill-posedness is identi, ed as one of the main characteristics of inverse problems. It is clear that if we have a noisy data, the inverse problem becomes unstable. As such, a numerical procedure based on discrete molli, cation and space marching method is applied to address the ill-posedness of the mentioned problem. The Regularization parameter is selected by generalized cross validation (GCV) method. The numerical stability and convergence of the proposed method are investigated. Finally, some test problems, whose exact solutions are known, are solved using this method to show the e, ciency.

In this paper, we propose an inexact alternating direction method with square quadratic proximal (SQP) Regularization for the structured variational inequalities. The predictor is obtained via solving SQP system approximately under signi cantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original SQP method. Under appropriate conditions, the global convergence of the proposed method is proved. We show the O(1=t) convergence rate for the inexact SQP alternating direction method. We also reported some numerical results to illustrate the e ciency of the proposed method.

One of the challenges of convolutional neural networks (CNNs), as the main tool of deep learning, is the large volume of some relevant models. CNNs, inspired form the brain, have millions of connections. Reducing the volume of these models is done by removing (pruning) the redundant connections of the model. Optimal Brain Damage (OBD) and Sparse Regularization are among the famous methods in this field. In this study, a deep learning model has been trained and the effect of reducing connections with the aforementioned methods on its performance has been investigated. As the proposed approach, by combining the OBD and Regularization methods its redundant connections were pruned. The resulting model is a smaller model, which has less memory and computational load than the original model, and at the same time its performance is not less than the original model. The experimental results show that the hybrid approach can be more efficient than each of the methods, in the most tested datasets. In one dataset , with the proposed method, the number of connections were reduced by 76%, without sacrificing the efficiency of the model. This reduction in model size has decreased the processing time by 66 percent. The smaller the software model, the more likely it is to be used on weaker hardware, found everywhere, and web applications.

In this paper Legendre wavelet bases have been used for finding approximate solutions to singular boundary value problems arising in physiology. When the number of basis functions are increased the algebraic system of equations would be ill-conditioned (because of the singularity), to overcome this for large M, we use some kind of Tikhonov Regularization. Examples from applied sciences are presented to demonstrate the efficiency and accuracy of the method.