Ill-conditioned linear systems may frequently arise in discretization of integral equations and many other real world applications. Solving such systems by classical methods might fail or result to solutions that are meaningless from practical point view. Moreover a slight perturbation in the right hand side vector might also lead to an enormous change of the solution vector due to ill-conditioned ness. To find meaningful solutions of such systems, the Tikhonov Regularization is an effective technique that has been widely used. In this paper we use it to solve ill-conditioned linear systems and also to find closest feasible linear systems to nearly feasible linear systems by smallest changes in problem data. Throughout the paper numerical results are reported to demonstrate the practical efficiency of the presented algorithms.

Super-resolution (SR) aims to overcome the ill-posed conditions of image acquisition. SR facilitates scene recognition from low-resolution image(s). Various approaches have tried to aggregate the informative details of multiple low-resolution images into a high-resolution one. In this paper, we present a new robust fuzzy super resolution approach. Our approach, firstly registers two input image using SIFT-BP-RANSAC registration. Secondly, due to the importance of information gain ratio in the SR outcomes, the fuzzy Regularization scheme uses the prior knowledge about the low-resolution image to add the amount of lost details of the input images to the registered one using the common linear observation model. Due to this fact, our approach iteratively tries to make a prediction of the high-resolution image based on the predefined Regularization rules. Afterwards the low-resolution image have made out of the new high-resolution image. Minimizing the difference between the resulted low-resolution image and the input low-resolution image will justify our Regularization rules. Flexible characteristics of fuzzy Regularization adaptively behave on edges, detailed segments, and flat regions of local segments within the image. General information gain ratio also should grow during the Regularization. Our fuzzy Regularization indicates independence from the acquisition model. Consequently, robustness of our method on different ill-posed capturing conditions and against registration error noise compensates the shortcomings of same Regularization approaches in the literature. Our final results show reduced aliasing achievements in comparison with similar recent state of the art works.

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.