Convergence Of Iterative Process Research Articles

Multi-target Localization Based on an Improved FOCUSS Algorithm with Missing Information

The situation where the observation stations or the sensors cannot fully detect the targets is called missing information, which makes it difficult to locate targets. Due to its sparsity, Compressed Sensing (CS) can be applied to multi-target localization with missing information. An improved Focal Undetermined System Solver (IFOCUSS) algorithm is proposed, by dynamically updating the regularized factor that has a significant impact on the convergence of iteration processes. What’s more, a re-weighting method is adopted for the weighting matrix to improve convergence property and accelerate convergence, and a new method of choosing correct items from a sparse estimation vector is proposed to improve the positioning accuracy. Several simulations corroborate its notable reconstruction performance, which is illustrated by correctly estimating five targets with only 11 sensors on a moving station, achieving better convergence at lower signal-to-noise ratios (such as 12 dB) with about half the run time of Regularized FOCUSS (RFOCUSS) and being able to estimate two targets even with missing information up to 60%. Under the same conditions of missing information, IFOCUSS can estimate more targets with smaller positioning errors than RFOCUSS, achieving better convergence performance by eliminating almost all interference terms.

An analytical approximate solution for stress and displacement around a shallow circular tunnel excavated under concave slope topography

For shallow buried tunnels under slope, overloading on the ground surface and unloading of excavation both significantly influence the stability of the tunnel surrounding rock and slope. This study proposes an analytical approximate solution for the stress and displacement around shallow buried circular tunnel under slope boundary. In derivation process, the original problem is divided into two models: Model A is a slope plane without any holes subjected to gravity and surcharge load along the ground surface, and Model B is a tunnel contained in a slope plane loaded by released stresses along the tunnel boundary. Furthermore, Model B is decomposed into another two models: Model B1 represents a slope without holes subjected to virtual surface force along the ground surface, and Model B2 is an infinite domain containing a tunnel loaded by another virtual surface force along its boundary. Two sets of virtual surface forces are then achieved by an iterative calculation. The solution, which strictly meets all stress boundary and compatibility conditions of the original problem, are ultimately solved by superposing the solution of Models A, B1 and B2, respectively obtained by the method proposed by the authors (Models A, B1) and complex variable theory (Model B2). The convergence of iteration process and good agreement between the derived and numerical results under the same assumptions verify the correctness of the proposed solution. Furthermore, a parametric investigation is performed to investigate the influence of surcharge load position, slope topography and tunnel position on the stress and displacement around the tunnel and near the ground. The proposed solution is of great value for the conceptual understanding of mechanical behavior of shallow tunneling in slope terrain and for verifying the numerical models. Moreover, the solution is useful for the preliminary design of tunnels at shallow depth under slope topography.

Optimization of technological regimes of the deisopentanizer column for the preparation of the feed stream of the low-temperature isomerization unit PGI-DIG/280-NK

Raw fractions used in isomerization plants contain isopentane, which is the target product, and its presence in the feed as “ballast” occupies the useful volume of the reactor. To optimize the composition of processed raw materials, a number of enterprises have included a deisopentanizer column in the technological scheme. The paper evaluates the possibility of optimizing the technological regimes and design parameters of a deisopentanizer for a low-temperature isomerization unit that does not have a column in the technological scheme for extracting isopentane from the composition of the feedstock. The studies were carried out using the Honeywell UniSim Design modeling system. The properties of the fraction components were calculated using the Peng-Robinson method. Calculations were made for two types of raw materials containing 8.7÷19.7% wt. isopentane. The performed calculations established that in order to achieve almost complete recovery of isopentane in the feed stream from the hydrotreating section, the distillation column should contain 63 theoretical plates, and the minimum reflux ratio should be Rmin = 12.3. At the same time, the presence of normal pentane in the distillate of the column will not exceed 0.07% wt., and in the bottom product, the content of isopentane will also not exceed 0.07% wt. The operation of a distillation column is simulated, containing 80 plates (the efficiency of the contact device is 0.8), a feed plate is 33. The feedstock consumption is 62550 kg/h, the temperature is 40°C and the pressure is 294.2 kPa. The convergence of iterative processes by the Honeywell UniSim Design environment was obtained using active specifications: the content of isopentane in the cube is 0.0007wt. share and pentane in the distillate 0.0007wt. share. The obtained technological modes provide almost complete extraction of isopentane from the composition of the raw material, namely by 99.37÷99.7%, while for industrial conditions it is essential to implement the calculated temperature profile along the height of the column. The possibility of using a column with the same diameter in its exhaustive and strengthening parts is shown.

Newton’s method and its visualization

Relevance of work. Mathematical modulation in various fields of science and technology often leads to nonlinear equations or systems of such equations. Far from always, these equations can be solved by exact methods. More often it is necessary to use approximate methods. One of the most popular of them is Newton's method. In modern works, Newton's method often serves as the basis for the development of new approximate methods that accelerate the convergence of iterative processes or are used to solve systems of large orders. The goal of the work. Visualize the work of the algorithm for solving the equation, as well as the system of equations according to Newton's method, so that the results of this work could be used when compiling electronic textbooks on the study of this method. Another goal is to study the method in the case when the system has several solutions; to study the possibility of using the method for equations with an infinite number of solutions.

Convergence of a Constrained Vector Extrapolation Scheme

Among extrapolation methods, Anderson acceleration (AA) is a popular technique for speeding up convergence of iterative processes toward their limit points. AA proceeds by extrapolating a better approximation of the limit using a weighted combination of previous iterates. Whereas AA was originally developed in the context of nonlinear integral equations, or to accelerate the convergence of iterative methods for solving linear systems, it is also used to extrapolate the solution of nonlinear systems. Simple additional stabilization strategies can be used in this context to control conditioning issues. In this work, we study a constrained vector extrapolation scheme based on an offline version of AA with fixed window size, for solving nonlinear systems arising in optimization problems, where the stabilization strategy consists in bounding the magnitude of the extrapolation weights. We provide explicit convergence bounds for this method and, as a by-product, upper bounds on a constrained version of the Chebyshev problem on polynomials.

Location, Separation and Approximation of Solutions for Quadratic Matrix Equations

In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This study allow us to determine domains of existence and uniqueness of solutions, and therefore to locate and separate the solutions. Another goal is to approximate a solution of the quadratic matrix equation. For this, we consider iterative processes of fixed point type. So, analyzing the convergence of these iterative processes of fixed point type, we locate, separate and approximate solutions of quadratic matrix equations.

The Influence of Gauge Conditions on Process and Result of Solving 3-D Problems of Computational Electromagnetism by the Finite Element Method

This article discusses the properties of the practical application results of the Coulomb and the “tree-cotree” gauge conditions for problems based on massive finite element 3-D-meshes (up to 12 million tetrahedrons) and with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${A}$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> formulation of the magnetic potential. The influence of the sort of gauging of the magnetic vector potential on the accuracy of the calculated solution of systems of linear algebraic equations (SLAEs) is shown by the example of magnetostatics and quasi-static (frequency domain) benchmark problems. The study was carried out for the cases of solving problems by iterative methods of the Krylov subspace and by direct methods. Iterative methods have shown the complexity of achieving the required solution accuracy with an increase in the number of finite elements and deterioration of the properties of the SLAE. The analysis of the dependence of the iterative process convergence and the required solution accuracy on the number of elements in the mesh and the type of gauging applied is accomplished. For direct methods for solving SLAEs, an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> estimate of the dependence of the value of the relative forward error on the presence and type of the gauge condition is performed. The possibility and efficiency of using block low-rank (BLR) factorization for solving SLAEs in the presence and absence of gauge conditions are considered. The conditions for using the gauging of the magnetic vector potential are shown, which are suitable for simplifying the process of preparing and solving numerical problems and increasing the accuracy of the calculated values of the magnetic vector potential in the frequency domain.

Convergence of Mann and Ishikawa Iterative Processes for Some Contractions in Convex Generalized Metric Space

In this paper, we consider a JS metric space endowed with convexity structure, which will allow us to examine and study convergence of Mann iteration and Ishikawa iteration for Banach type and Chatterjea type contractions defined on JS metric space.

Оценки сходимости некоторых итерационных алгоритмов численного моделирования двумерных уравнений магнитной гидродинамики

In this work, the convergence of the methods of separate and combined solution of difference equations groups separated by physical processes, applied to completely conservative difference schemes (CCDS) of two-dimensional magnetohydrodynamics, is studied. The corresponding estimates for the convergence of iterative processes for the entire family of CCDS are obtained. The area of applicability of this method is shown. The validity of the estimates obtained in the work is confirmed by numerical calculations. Depending on the velocity of convergence of the iterative process, based on the estimates determined in this work, recommendations were developed on the use of one of the two methods under study (separate or combined solution of groups of difference equations).

Analysis of the most common methods for determining the stability of energy systems

Problem. There are many methods for determining the stability of the energy system. In normal operating condition (normal rated mode), the power system must reliably ensure the consumption of electricity of normalized quality. However, in addition to the normal state, there are emergency and transient states caused by various transients. This is due to the fact that the energy system is constantly changing its parameters. Such changes are determined by variations in the amount of power produced and consumed, as well as the changes in system configuration. Goal. The goal is studying the possibilities of various methods of determining the power systems stability and drawing up the general algorithm of actions for maintenance of their stability. Methodology. When determining the stability of energy systems by the Lyapunov method, two methods can be used: the direct method and the first approximation method. Lyapunov direct method refers to differential methods. To conclude about the stability of the system we do not find a general or particular solution of differential equations, but with their help we find a mathematical function, the complete derivative of which over time allows to obtain a conclusion about the stability of the system. Results. Many methods can be used to determine whether a sustainable energy system is stable or not. The most common are the Lyapunov methods and the Moiseev method. It is determined that the direct Lyapunov method refers to differential methods. The application of the direct Lyapunov method for energy problems is limited. Currently, it can be used only for some individual cases. The method of the first approximation (Lyapunov first method) has received wider application in the solution of power problems. When applying this method, which belongs to the group of methods of full integration, the right-hand sides of the equations are decomposed into power series. Originality. It is determined that one of the perspective directions of increasing the efficiency of the mathematical device work is using the methods of the second order in modeling and optimization of operating modes of electric power systems. This allows you to increase the speed and reliability of the convergence of iterative processes. Practical value. Based on the analysis of various existing methods for solving the problems of stability of energy systems, an algorithm of actions is proposed and developed, which will help to solve the problem of stability in practice.

Extended convergence analysis of Newton-Potra solver for equations

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator.

&lt;span&gt;Convergence of mann iteration process to a fixed point of (&lt;em&gt;&amp;alpha;&lt;/em&gt;&lt;em&gt;,&lt;/em&gt;&lt;em&gt;&amp;beta;&lt;/em&gt;) - nonexpansive mappings in Lp spaces&lt;/span&gt;

In this paper, we prove the convergence of Mann iteration to fixed points of (α,β) -nonexpansive and strictly pseudo-contractive mappings in spaces. In addition, by using the obtained results, we state the convergence of Mann iteration to solutions of the nonlinear integral equations.

Kirk type iteration process and convex feasibility problem in CAT(0) space

In this paper, we establish $$\bigtriangleup $$ -convergence and strong convergence of Kirk type iteration process in CAT(0) spaces. We apply our results to solve consistent and inconsistent convex feasibility problems in CAT(0) spaces. This work improves and extends the corresponding results of Kirk, Kuhfittig, Rhoades and Takahashi and Tamura on a nonlinear domain.

In-situ deflectometric measurement of optical surfaces for precision manufacturing

The in-situ measurement of complex optical surfaces is a challenging task in precision engineering. The phase measuring deflectometry is a powerful measuring method for complex specular surfaces, and it has higher measuring efficiency, stability and dynamic range compared to interferometry. Consequently it is promising to widespread applications in various fields. Deflectometry is essentially a calibration problem, and the measuring accuracy is directly determined by the quality of geometrical calibration. An in-situ deflectometric measuring system is designed based on the single point diamond turning machine. A self-calibration method is developed to specify the relative positions of the camera and screen. Ray tracing is conducted at two positions of an auxiliary reflecting mirror, which is mounted on an air bearing spindle. The accuracy of the geometrical positions can be improved by an order of magnitude by minimizing the deviations of the traced points with respect to the true correspondences. According to the statistical properties of the deviations in reverse ray tracing, the form errors and the position errors can be separated, and the positioning error of the workpiece can be corrected accordingly. Henceforth, the nominal shape of the fabricated workpiece can be fully utilized, and the conventional one-way position-form mapping can be converted into a two-way mapping problem. As for the complex shapes, the whole surface can be covered by sub-aperture measurement. Precise localization of a local region under test is achieved by multi-position imaging, so that correct convergence of the iterative reconstruction process can be guaranteed. Several typical optical surfaces including an off-axis paraboloid mirror are measured, and the measuring accuracy of the proposed method is proved better than 150 nm RMS.

Convergence of a new three-step iteration process to common fixed points of three G-nonexpansive mappings in Banach spaces with directed graphs

In this paper, from some confusions in examples on the convergence of certain iteration processes in the literature, we introduce the coordinate-convexity to study the convergence of a new three-step iteration process in Banach spaces with directed graphs. We prove some comparison results of the rate of convergence of the proposed iteration process and certain iteration processes for G-contractive mappings. We also prove the weak and strong convergence results for three G-nonexpansive mappings, and give three numerical examples to illustrate the obtained results.

A discrete Schwarz waveform relaxation method of higher order for singularly perturbed parabolic reaction–diffusion problems

We consider singularly perturbed parabolic reaction–diffusion problems of scalar and vector types. We construct a discrete Schwarz waveform relaxation method of higher order for their numerical solution. The method is shown to be parameter-uniformly convergent having almost fourth order in space and first order in time. Further, interesting result proven is much faster convergence of iterative process for small perturbation parameter. Numerical results demonstrating the efficiency of the proposed method and validating the theoretically proven convergence results are given.

A convergence study of monolithic simulations of flow and deformation in fractured poroelastic media

SummaryA consistent linearisation has been carried out for a monolithic solution procedure of a poroelastic medium with fluid‐transporting fractures, including a comprehensive assessment of the convergence behaviour. The fracture has been modelled using a subgrid scale model with a continuous pressure across the fracture. The contributions to the tangential stiffness matrix of the fracture have been investigated to assess their impact on convergence. Simulations have been carried out for different interpolation orders and for Non‐Uniform Rational B‐Splines as interpolants vs Lagrangian polynomials. To increase the generality of the results, Newtonian as well as non‐Newtonian (power‐law) fluids have been considered. Unsurprisingly, a consistent linearisation invariably yields a quadratic convergence but comes at the expense of a loss of symmetry and recalculation of the contribution of the interface to the stiffness matrix at each iteration. When using a linear line search, however, the inclusion of only those terms of the interface stiffness that result in a symmetric and constant tangential stiffness matrix is sufficient to obtain a stable and convergent iterative process.

Convergence of the Ishikawa Type Iteration Process with Errors of I-Asymptotically Quasi-nonexpansive Mappings in Cone Metric Spaces

The goal of this article is to consider an Ishikawa type iteration process with errors to approximate the fixed point of -asymptotically quasi non-expansive mapping in convex cone metric spaces. Our results extend and generalize many known results from complete generalized convex metric spaces to cone metric spaces.

Meshless local Petrov Galerkin method for 2D/3D nonlinear convection–diffusion equations based on LS-RBF-PUM

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear convection–diffusion equations. The proposed method is not sensitive to the node layout, and has good stability and flexibility to complex domain. In order to treat nonlinear term, Picard iterative scheme is employed to confirm the convergence of iterative process. Error estimates are derived by the radial basis function interpolation method and convergence rate is proven to be second order. Numerical examples are performed for the nonlinear convection–diffusion equations in two and three space dimensions (2D/3D), which not only supports the theoretical results but also finds out superconvergence of third order.

STUDY OF PLASTIC FORMING IN PRODUCTION OF THERMOELECTRIC BISMUTH TELLURIDE BASED MATERIAL

An experimental and theoretical study of the process of the equal−channel angular pressing (ECAP) was performed to obtain a thermoelectric (TE) material based on bismuth telluride. A brief review of the mathematical modeling of the ECAP process is given. The influence of the ECAP design features and temperature modes on the process of plastic forming is considered. The results of calculations of the thermally stressed state of samples at different stages of the ECAP process are presented. The calculations of the ECAP process were carried out by means of Lagrangian finite element mesh, which adjusted adaptively during the process to the die geometry and became finer or coarser depending on the magnitude of the plastic deformation. It was required for the specified calculation accuracy and the convergence of iterative process. The results of an experimental study of the structure and properties of samples obtained by the ECAP using a set of measuring methods (X−ray diffractometry and electron microscopy) are discussed. The thermoelectric characteristics of the obtained materials were measured by Harman method. Comparative methodical calculations of the ECAP process for TE materials based on bismuth telluride have been made by adjusting parameters determining the grain formation (i.e. the critical plastic deformation as a function of temperature and power−law dependence of its rates). It made possible to adjust the ECAP model on the basis of the measured grain sizes for TE materials . The calculation results of grain creation during the plastic forming, which are compared with the measurement data, are presented. The practical result of this research was the improved geometry of the die and the validated technological regimes of plastic deformation, which allowed obtaining samples with the good TE efficiency.