The main purpose of this article is to increase the efficiency of the LEAST SQUARES method in numerical solution of ill-posed functional and physical equations. Determining the LEAST SQUARES of a given function in an arbitrary set is often an ill-posed problem. In this article, by defining artificial constraint and using Lagrange multipliers method, the attempt is to turn n-dimensional LEAST SQUARES PROBLEMS into (n-1) ones, in a way that the condition number of the corresponding system with(n-1) -dimensional problem will be low. At first, the new method is introduced for2 and 3-term basis, then the presented method is generalized for n-term basis. Finally, the numerical solution of some ill-posed PROBLEMS like Fredholm integral equations of the first kind and singularly perturbed linear Fredholm integral equations of the second kind are approximated by chain LEAST SQUARES method. Numerical comparisons indicate that the chain LEAST SQUARES method yields accurate and stable approximations in many cases.

The Mixed LEAST SQUARES Meshfree (MDLSM) method has shown its appropriate efficiency for solving Partial Differential Equations (PDEs) governing the engineering PROBLEMS. The method is based on the minimizing the residual functional. The residual functional is defined as a summation of the weighted residuals on the governing PDEs and the boundaries. The Moving LEAST SQUARES (MLS) is usually applied in the MDLSM method for constructing the shape functions. Although the required consistency and compatibility for the approximation function is satisfied by the MLS, the method loss its appropriate efficiency when the nodal points cluster too much. In the current study, the mentioned drawback is overcome using the novel approximation function called Mapped Moving LEAST SQUARES (MMLS). In this approach, the cluster of closed nodal points maps to standard nodal distribution. Then the approximation function and its derivatives compute noting the some consideration. The efficiency of suggested MMLS for overcoming the drawback of MLS is evaluated by approximating the mathematical function. The obtained results show the ability of suggested MMLS method to solve the drawback. The suggested approximation function is applied in MDLSM method, and used for solving the Burgers equations. Obtained results approve the efficiency of suggested method.

Meshless methods have been added to numerical methods in recent decades, and have provided a wide range of scientific, research and engineering fields. The use of Meshless methods is still not extent to the finite element methods in engineering issues, but these methods may now be similar to those of the time when the finite element method begins to expand. In this research, a discrete LEAST square meshless method with collocation points CDLSM is proposed. The concepts, mathematical relations, and formulation of this method are fully presented. In this simulation, collocation points are used for more efficiency and lower computing time by using LEAST SQUARES method, as well as using the series instead of integrals (discrete mode). Based on this method, the dam failure phenomenon has been solved in different cases and its verification has been used by comparison with analytical solution with experimental data whenever it is available. Comparison of numerical results with existing analytical and experimental data shows that the method has high efficiency and simulates the shock or discontinuity.

This study compared the performance of the partial LEAST SQUARES-structural equation modelling (PLS-SEM) and the robust partial LEAST SQUARES-structural equation modelling (RPLS-SEM) methods through Winsorisation approach The inputs and the outputs used in this model were based on the electricity generation data, derived from the Al-Zawiya Steam Power Plant, Libya. Furthermore, the researchers compared the novel RPLS-SEM approach with the traditional PLS-SEM approach and noted that the novel RPLS-SEM method was more efficient compared to PLS-SEM.

In this paper, we propose a nonparametric rank-based alternative to the LEAST-SQUARES independent component analysis algorithm developed. The basic idea is to estimate the squared-loss mutual information, which used as the objective function of the algorithm, based on its copula density version. Therefore, no marginal densities have to be estimated. We provide empirical evaluation of the proposed algorithm through simulation and real data analysis. Since the proposed algorithm uses rank values rather than the actual values of the observations, it is extremely robust to the outliers and suffers less from the presence of noise than the other algorithms.

Determination of the diffusion coefficient on the base of solution of a linear inverse problem of the parameter estimation using the LEAST-square method is presented in this research. For this propose a set of temperature measurements at a single sensor location inside the heat conducting body was considered. The corresponding direct problem was then solved by the application of the heat fundamental solution.