The method of detection and localization of configuration defects in geodetic networks by means of Tikhonov regularization

In adjusted geodetic networks, cases of local configuration defects (defects in the geometric structure of the network due to missing data or errors in point numbering) can be encountered, which lead to the singularity of the normal equation system in the least-squares procedure. Numbering errors in observation sets cause the computer program to define the network geometry incorrectly. Another cause of a defect may be accidental omission of certain data records, causing local indeterminacy or lowering of local reliability rates in a network. Obviously, the problem of a configuration defect may be easily detectable in networks with a small number of points. However, it becomes a real problem in large networks, where manual checking of all data becomes a very expensive task. The paper presents a new strategy for the detection of configuration defects with the use of the Tikhonov regularization method. The method was implemented in 1992 in the GEONET system (www.geonet.net.pl).

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

USING A DOUBLE CONVOLUTION OF LORENTZ AND GAUSS FUNCTIONS FOR PROCESSING THE MÖSSBAUER SPECTRA OF THE SUPERSATURATED DISORDERED SOLID SOLUTIONS

An algorithm for mathematical processing of the Mössbauer spectra of supersaturated disordered solid solutions by the Tikhonov regularization method using a double convolution of the Lorentz function and two Gaussians is proposed. By the examples of spectra of supersaturated disordered solid solutions Fe100–xGex (x = 10—25 at.%) and Fe75Si15Al10, it is shown that the algorithm allows more correct processing, which provides a reliable distribution function of the hyperfine magnetic field. It is shown that to take into account the statistical ensemble of nonequivalent local atomic configurations of Fe atoms in disordered supersaturated solid solutions, it is necessary to use not only the convolution of two Gaussian functions, but also the projection scaling factor of the hyperfine magnetic field onto the velocity scale.

Reliability Analysis of Intelligent Electric Energy Meter under Fusion Model Illness Analysis Algorithm

This work is aimed at solving the morbidity problem of the smart meter fusion model and improve the measurement accuracy and reliability of the smart meter. Starting with the topology of the smart meter, the reason for the serious morbidity of the smart meter model is discussed. First, the basic process of power system state estimation of smart meters is introduced, and the concept of error analysis of smart meters is clarified. Then, the causes and mechanisms of the ill-conditioned problems of the smart meter model are analyzed, and methods to reduce the morbidity of the smart meter calculation model are analyzed. Finally, a data optimization algorithm based on a greedy strategy and an improved Tikhonov regularization method is proposed. The model data is processed and optimized to reduce the morbidity of the smart meter measurement model. The results show that the analysis algorithm for reducing the morbidity error of the smart meter proposed in this study can effectively interfere with the morbidity of the smart meter calculation model. The processing effect shows that it can reduce the measurement error of the smart meter to about 5%, which is an order of magnitude lower than the error before processing, and the processing effect of the least square method is improved by more than 70%. From the perspective of processing speed, when the user number is between 50 and 100, the running time of the algorithm ranges between 1.5 and 3.5 s, which can be fully adapted to the actual situation and has strong practicability. In short, this study is helpful in improving the accuracy and reliability of smart meter calculations and provides a certain reference for related research.

An Efficient Algorithm for Ill-Conditioned Separable Nonlinear Least Squares

For separable nonlinear least squares models, a variable projection algorithm based on matrix factorization is studied, and the ill-conditioning of the model parameters is considered in the specific solution process of the model. When the linear parameters are estimated, the Tikhonov regularization method is used to solve the ill-conditioned problems. When the nonlinear parameters are estimated, the QR decomposition, Gram–Schmidt orthogonalization decomposition, and SVD are applied in the Jacobian matrix. These methods are then compared with the method in which the variables are not separated. Numerical experiments are performed using RBF neural network data, and the experimental results are analyzed in terms of both qualitative and quantitative indicators. The results show that the proposed algorithms are effective and robust.

Comparison of Algebraic Reconstruction Technique Methods and Generative Adversarial Network in Image Reconstruction of Magnetic Induction Tomography (MIT)

Magnetic induction tomography (MIT) is a technique used for imaging electromagnetic properties of objects using eddy current effects. The non-linear characteristics had led to more difficulties with its solution especially in dealing with low conductivity imaging materials such as biological tissues. Two methods that could be applied for MIT image processing which is the Generative Adversarial Network (GAN) and the Algebraic Reconstruction Technique (ART). ART is widely used in the industry due to its ability to improve the quality of the reconstructed image at a high scanning speed. GAN is an intelligent method which would be able to carry out the training process. In the GAN method, the MIT principle is used to find the optimum global conductivity distribution and it is described as a training process and later, reconstructed by a generator. The output is an approximate reconstruction of the distribution’s internal conductivity image. Then, the results were compared with the previous traditional algorithm, namely the regularization algorithm of BPNN and Tikhonov Regularization method. It turned out that GAN had able to adjust the non-linear relationship between input and output. GAN was also able to solve non-linear problems that cannot be solved in the previous traditional algorithms, namely Back Propagation Neural Network (BPNN) and Tikhonov Regularization method. There are several other intelligent algorithms such as CNN (Convolution Neural Network) and K-NN (K-Nearest Neighbor), but such algorithms have not been able to produce the expected image quality. Thus, further study is still needed for the improvement of the image quality. The expected result in this study is the comparison of these two techniques, namely ART and GAN to get the best results on the image reconstruction using MIT. Thus, it is shown that GAN is a better candidate for this purpose.

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

In this paper, the numerical analytic continuation problem is addressed and a fractional Tikhonov regularization method is proposed. The fractional Tikhonov regularization not only overcomes the difficulty of analyzing the ill-posedness of the continuation problem but also obtains a more accurate numerical result for the discontinuity of solution. This article mainly discusses the a posteriori parameter selection rules of the fractional Tikhonov regularization method, and an error estimate is given. Furthermore, numerical results show that the proposed method works effectively.