Use Of Iterative Methods Research Articles

Plug-and-Play Image Reconstruction Is a Convergent Regularization Method.

Non-uniqueness and instability are characteristic features of image reconstruction methods. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides a family of stable reconstructions that converge to a specific solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by a variational image reconstruction that minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the regularizer. In recent years, Plug-and-Play (PnP) image reconstruction has been developed as a new powerful generalization of variational methods based on replacing proximal mappings by more general image denoisers. While PnP iterations yield excellent results, neither stability nor convergence in the sense of regularization have been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent regularization methods. This shows for the first time that PnP is as mathematically justified for robust image reconstruction as variational methods.

SOLUTION OF PHYSICALLY NONLINEAR PROBLEMS OF DEFORMATION OF MASSIVE AND THIN-WALLED PRISMATIC BODIES

The results of a numerical study on test examples of the convergence of the developed version of the semi-analytical method of finite elementsare presented. A comparison of machine time and accuracy of solving elastic and plastic problems obtained on the basis of finite elements with variableand averaged mechanical parameters was performed.The convergence of the finite element method and the semianalytical finite element method in the calculation of prismatic bodies loaded with localized and distributed external influences was studied. An effective numerical approach to the study of arbitrarily loaded massive and thin-walled prismatic bodies of complex shape, the deformation of which can pass beyond theelastic limit of the material, is being developed based on the finite element moment scheme and the semianalytical variant of the finite element method. Due to the representation of displacements by polynomials and the use of iterative methods of solving systems of equations, this approach is developed in relation to the calculation of objects with arbitrary boundary conditions on the ends, which made it possible to expand the area of effective application of the semi-analytical method of finite elements to a new class ofproblems.A number of new complex problems of elastic and elastic-plastic deformation of massive and thinwalled prismatic bodies, which have an independentapplied value, have been solved.

Fast successive permutation iterative algorithms for solving convection diffusion equation

The solution of convection diffusion problem is a complex task, requiring the solving of linear equations which can take up a significant amount of computational time. This issue can be mitigated through the use of iterative methods. We present a class of fast successive permutation iterative algorithms with relaxation factors (ω-SPI) to solve 1D, 2D and 3D convection diffusion equations. The stability, convergence and optimal value of the relaxation factor are proved. In order to test the efficiency and accuracy of our proposed iterative algorithms, we present several numerical experiments. The results demonstrate that the new algorithms are more accurate and faster than traditional methods such as full-implicit, Gauss-Seidel and SOR algorithms.

LEAP – A Process for Identifying Potential Technical Debt in Iterative System Development

AbstractSystems engineering has seen a rise in the use of iterative methods to design and develop both hardware and software systems which allow for system refinement to be responsive to user needs. However, focusing on items with high value to the user can result in technical debt, where technical compromises made for short‐term gain impact the long‐term health of the system. Current methods for identifying technical debt focus on finding existing technical debt items within a system and not on proactive identification of technical debt during the iterative system planning process. This paper presents a novel technique to identify technologies that impact the ability of the system to satisfy the needs of its stakeholders. The method is used to evaluate different choices of technological implementations in both the temporal and the functional dimensions to reduce the risk of incurring technical debt which prevents the successful delivery of the system.

A new approach to numerical simulation of boundary value problems of the theory of elasticity in stresses and strains

The main parameters characterizing the process of deformation of solids are displacements, strain and stress tensors. From the point of view of the strength and reliability of the structure and its elements, researchers and engineers are mainly interested in the distribution of stresses in the objects under study. Unfortunately, all boundary value problems are formulated and solved in solid mechanics mainly with respect to displacements, or an additional stress functions. And the required stresses are calculated from known displacements or stress functions. In this case, the accuracy of stress calculation is strongly affected by the error of numerical differentiation, as well as the approximation order of the boundary conditions. The formulation of boundary value problems directly with respect to stresses or strains allows to increase the accuracy of stress calculation by bypassing the process of numerical differentiation.
 Therefore, the present work is devoted to the formulation and numerical solution of boundary value problems directly with respect to stresses and strains. Using the well-known Beltrami-Miеchell equation, and considering the equilibrium equation as ah additional boundary condition, a boundary value problem(BVP) is formulated directly with respect to stresses. In a similar way, using the strain compatibility condition, the Beltrami-Mitchell type equations for strains are written.
 The finite difference equations for two-dimensional BVP are constructed and written in convenient a form for the use of iterative method. A number of problems on the equilibrium of a rectangular plate under the action of various loads applied on opposite sides are numerically solved. The reliability of the results is ensured by comparing the numerical results of the 2D elasticity problems in stresses and strains, and with the exact solution, as well as with the known solutions of the plate tension problem under parabolic and uniformly distributed loads

STUDY OF THE CONDENSER’S OPERATING CHARACTERISTICS OF THE NPP UNIT

The paper shows that the technical condition of condensing devices of steam turbines largely determines the amount of electricity losses, reliable and economical operation of NPP units, and the modernization of steam turbine capacitors will provide a significant increase in electricity generation with relatively low capital investment compared to construction of new NPP power units. Analysis of the perfection of the heat transfer process in the condenser according to the principles of thermoeconomic diagnostics to identify the causes of abnormal operation of the power conversion system showed that the main causes of load reduction are determined by rising cooling water temperature and deviation of steam pressure from normal assessment of contamination of the heat transfer surface, which significantly affects the reduction of electricity generation. The modernization main points of the capacitor of the Zaporizhzhya NPP power unit № 3 on the principle of "block-modular" design developed by PJSC "Turboatom" and the characteristics of the capacitor provided by the new design are considered. The method of thermal calculation of the condenser with the use of iterative methods, which takes into account the presence of air leakage in the space of the condenser and the appearance of contamination of the heat exchange surface, is given. The increase of electric power at the generator terminals at change of steam load factor and pollution thickness is calculated and analyzed. An analytical dependence of the investigated parameters of reducing the increase in electric power during the modernization of the capacitor is obtained and the interpretation of the results is carried out. The work emphasizes the importance of processing the results of diagnostics and monitoring of the technical condition of steam turbines condensing devices and information about their impact on the amount of electric power generation.

Railway track deflection analysis by using evolutionary algorithms

In contrast to numerical methods, analytical modelling of the railway track is one of the less time-consuming and computationally demanding methods which, in combination with the computing power commonly available today, can form an effective tool for analysing the behaviour of the railway track.This paper deals with the use of iterative methods of evolutionary algorithms, together with analytical modelling, for the purpose of reverse analysis of the measured deflection caused by moving loads acting on the railway track. The theoretical assumptions of the analytical model used, the data collection methodology and the method used for the reverse analysis are presented. The results of the analysis are also presented.

The Smallest Singular Value Anomaly and the Condition Number Anomaly

Let A be an arbitrary matrix in which the number of rows, m, is considerably larger than the number of columns, n. Let the submatrix Ai,i=1,…,m, be composed of the first i rows of A. Let βi denote the smallest singular value of Ai, and let ki denote the condition number of Ai. In this paper, we examine the behavior of the sequences β1,…,βm, and k1,…,km. The behavior of the smallest singular values sequence is somewhat surprising. The first part of this sequence, β1,…,βn, is descending, while the second part, βn,…,βm, is ascending. This phenomenon is called “the smallest singular value anomaly”. The sequence of the condition numbers has a different character. The first part of this sequence, k1,…,kn, always ascends toward kn, which can be very large. The condition number anomaly occurs when the second part, kn,…,km, descends toward a value of km, which is considerably smaller than kn. It is shown that this is likely to happen whenever the rows of A satisfy two conditions: all the rows are about the same size, and the directions of the rows scatter in some random way. These conditions hold in a wide range of random matrices, as well as other types of matrices. The practical importance of these phenomena lies in the use of iterative methods for solving large linear systems, since several iterative solvers have the property that a large condition number results in a slow rate of convergence, while a small condition number yields fast convergence. Consequently, a condition number anomaly leads to a similar anomaly in the number of iterations. The paper ends with numerical experiments that illustrate the above phenomena.

An Algorithm Derivative-Free to Improve the Steffensen-Type Methods

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.

Enhancing Accuracy Determination of Sources of Radio Emission Coordinates by Linear Antenna Arrays

Two methods have been proposed for enhancing the accuracy in determining the coordinates of radio emission sources by using phase systems based on reducing the dynamic and random errors in terms of the curvature of electromagnetic wave phase front during processing the signals at inputs of linear antenna array. The reduction of dynamic errors stipulated by the limited number of terms of the Maclaurin expansion of the distance-to-source function is achieved by estimating each value of distance and bearing without use of iterative methods. The reduction of random errors occurring during the source motion is performed at the expense of preliminary estimation of values of phase difference of received signals in the process of bearing estimation and values of the difference of phase differences in determining the distance by using the least squares method in the sliding window.

An Analytic Expression for the Inverse Involute

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

A Noniterative Method for Solving Nonlinear Equations for Steady-State Regimes of Electrical Networks

Methods for solving linear and nonlinear equations for steady-state operation modes are important for the development and operation of electrical networks. Linear equations are solved analytically or with the use of iterative methods, while nonlinear equations are solved only with use of iterative methods, usually the Newton–Raphson method. Iterative methods have two significant drawbacks. First, the iterative process can diverge, with the divergence being dependent on both the form of the nonlinear equation and the choice of the initial approximation. Second, in the case of convergence, the iterative method allows finding only one solution for each initial approximation, whereas a nonlinear equation can have several solutions corresponding to different operation modes of the electric network. Development of methods that are free of these drawbacks is an important problem. This paper proposes a noniterative method for solving nonlinear equations describing the steady-state operation modes of electric networks; the equations are written in the complex form. The method uses the resultant to transform set of nonlinear equations to a set of polynomials. The polynomials are composed in such a way that their zeros determine one of the coordinates of the solution vector of the nonlinear equation system. Thus, the problem of solving of the nonlinear equations system is reduced to the well-studied problem of finding the zeros of one-variable polynomials. There are numerous methods of solving this problem that do not require specifying the initial approximation and allow finding all zeros. The effectiveness of the method is illustrated by solving the equations of node network voltages in the form of power balance. The equations were solved by the traditional iterative method and the proposed method using the MathCAD-15 program. The drawbacks of the iterative method are illustrated by its divergence at certain values of the initial approximation and by the failure of finding all the solutions. In contrast, the use of the proposed noniterative method allows one to find two solutions and to establish that there is no other solutions. Further development of this method will be connected with taking into account the sparseness of the matrix of conductivities that would allow reducing the degree used polynomials, as well as with the extension of this method to the domain of real values in which the equations can be written down in the algebraic and trigonometric form.

Iterative Solvers for Empirical Partial Atomic Charges: Breaking the Curse of Cubic Numerical Complexity.

Rational drug design involves a vast amount of computations to get thermodynamically reliable results and often relies on atomic charges as a means to model electrostatic interactions within the system. Computational inefficiency often hampers the development of new and wider dissemination of the known methods; thus, any source to speed up the calculations without a sacrifice in quality is warranted. At the heart of many empirical methods of calculating atomic charges is the solution of a system of linear algebraic equations (SLAE). The classical method of solving SLAE-the Gauss method-has in general case a cubic computational complexity. It is shown that the use of iterative methods for solving SLAE, characteristic to typical empirical atomic charge calculation methods, makes it possible to significantly reduce the amount of calculations and to obtain a computational complexity approaching a quadratic one. Despite the fact that this phenomenon is well-known in numerical methods, iterative solvers surprisingly do not seem to have been systematically applied to calculation of atomic charges via empirical schemes. Another finding is the relative values of the matrix elements, determined by the physical grounds of the interactions within the empirical system, generally lead to SLAE's with well-defined matrices, suited to use with iterative solvers to fasten computation compared to using the noniterative solvers. This finding broadens the applicability range of atomic charges obtained with empirical methods for such cases as, e.g., account of polarizability via "on-the-fly" recalculation of charges in changing surroundings within the force fields in molecular dynamics settings.

Linear power flow formulation for low-voltage DC power grids

This paper presents a reformulation of the power flow problem in low-voltage dc (LVDC) power grids via Taylor's series expansion. The solution of the original nonlinear quadratic model is achieved with this proposed formulation with minimal error when the dc network has a well defined operative conditions. The proposed approach provides an explicit solution of the power flow equations system, which avoids the use of iterative methods. Such a characteristic enables to provide accurate results with very short processing times when real operating scenarios of dc power grids are analyzed. Simulation results verify the precision and speed of the proposed method in comparison to classical numerical methods for both radial and mesh configurations. Those simulations were performed using C++ and MATLAB, which are programming environments commonly adopted to solve power flows.

Plenoptic Image Motion Deblurring.

We propose a method to remove motion blur in a single light field captured with a moving plenoptic camera. Since motion is unknown, we resort to a blind deconvolution formulation, where one aims to identify both the blur point spread function and the latent sharp image. Even in the absence of motion, light field images captured by a plenoptic camera are affected by a non-trivial combination of both aliasing and defocus, which depends on the 3D geometry of the scene. Therefore, motion deblurring algorithms designed for standard cameras are not directly applicable. Moreover, many state of the art blind deconvolution algorithms are based on iterative schemes, where blurry images are synthesized through the imaging model. However, current imaging models for plenoptic images are impractical due to their high dimensionality. We observe that plenoptic cameras introduce periodic patterns that can be exploited to obtain highly parallelizable numerical schemes to synthesize images. These schemes allow extremely efficient GPU implementations that enable the use of iterative methods. We can then cast blind deconvolution of a blurry light field image as a regularized energy minimization to recover a sharp high-resolution scene texture and the camera motion. Furthermore, the proposed formulation can handle non-uniform motion blur due to camera shake as demonstrated on both synthetic and real light field data.

ПРОБЛЕМИ ВИБОРУ ПРІОРИТЕТНОГО НАПРЯМКУ СТРАТЕГІЧНОГО РОЗВИТКУ ПІДПРИЄМСТВА

In the article the choice of strategic priorities of the company. The use of iterative method to solve the problem of managing complex processes, providing choice as the starting point of the first iteration of the environment, since the choice of strategic priorities of enterprise development begins with an analysis of changes in the environment. That it creates a need and demand for innovation of a degree of aggressiveness. Quantitatively integral indicator for assessing the degree of aggressiveness innovation strategy defined by increments of adaptation costs and share new products. This kind of multiplier degree of aggressiveness innovation strategy. The higher the value of the multiplier, the higher the degree of aggressiveness innovation strategy. To implement the reduced model developed by an iterative process of selecting the best alternative strategic development of the company, which involves three iterations, during which gradually refined features, structures, processes, state of the environment, the implementation of which leads to the adoption of the necessary decisions. The major problem with this choice is the criterion of efficiency options strategic direction of the company. It is proposed as a criterion for selecting the best option to take rate of return on investment. The algorithm allows an iterative process of selecting priority strategic enterprise development in interactive mode. The algorithm consists of two parts: the first evaluates the degree of aggressiveness of innovation; second - choice of strategic direction of the company. The logic of the first part of the algorithm based on an assessment of aggressiveness innovation. The logic of the second part of the algorithm - is an iterative approach to the choice of strategic direction. In addition, the choice of strategic direction and development company based on an evaluation of elements of the strategic potential of the company.

The Use of Iterative Methods in Solving Tasks of Structural Mechanics

The article discusses some aspects of use of iterative methods for solving nonlinear tasks in structural mechanics. The example demonstrates the ability of the method of additional loads applied to various models of continuous media. Possibilities of the additional loads method in relation to different sections of structural mechanics by using few examples are also demonstrated. The advantages of the used method are discussed.

Scalable high-dimensional dynamic stochastic economic modeling

We present a highly parallelizable and flexible computational method to solve high-dimensional stochastic dynamic economic models. Solving such models often requires the use of iterative methods, like time iteration or dynamic programming. By exploiting the generic iterative structure of this broad class of economic problems, we propose a parallelization scheme that favors hybrid massively parallel computer architectures. Within a parallel nonlinear time iteration framework, we interpolate policy functions partially on GPUs using an adaptive sparse grid algorithm with piecewise linear hierarchical basis functions. GPUs accelerate this part of the computation one order of magnitude thus reducing overall computation time by 50%. The developments in this paper include the use of a fully adaptive sparse grid algorithm and the use of a mixed MPI-Intel TBB-CUDA/Thrust implementation to improve the interprocess communication strategy on massively parallel architectures. Numerical experiments on “Piz Daint” (Cray XC30) at the Swiss National Supercomputing Centre show that high-dimensional international real business cycle models can be efficiently solved in parallel. To the best of our knowledge, this performance on a massively parallel petascale architecture for such nonlinear high-dimensional economic models has not been possible prior to present work.

Special Section: 2014 Copper Mountain Conference

The Thirteenth Copper Mountain Conference on Iterative Methods was held April 6--11, 2014, in Copper Mountain, Colorado. The meeting featured approximately 130 talks covering a wide range of topics in scientific computing, including multigrid methods, domain decomposition, Krylov and Anderson accelerators, hybrid direct/iterative methods, iterative regularization of inverse problems, nonlinear solvers, algorithms for eigenvalue computations, and iterative methods in optimization. Numerous talks dealt with applications of iterative solvers to fields such as neutronics, image processing, graph mining, quantum chemistry, fluid flow, and wave propagation. Several sessions featured work on the use of iterative methods for solving coupled multiphysics problems arising in magnetohydrodynamics, atmospheric and ocean modeling, continuum mechanics, and so forth. Other presentations offered an overview of recent progress on resilient iterative methods, including work on stochastic solvers and fault-tolerant multigrid variants. As in the past, there were additional talks addressing various discretization methods for partial differential equations and their impact on the performance of iterative solvers. An encouraging sign for the future of the field was the strong student presence, accounting for over a third of the attendees. One of the high points of the Copper Mountain Conference is the traditional Student Paper Award Competition, which this year received an especially high number of very strong submissions. The Student Paper Award Committee charged with the selection of the best student papers had a very difficult job to do but eventually identified four entries deserving of recognition. A plenary session was devoted to the presentations by the four student winners: Jessica Bosch from the Max Planck Institute in Magdeburg, Germany (first place), Sarah Gaaf from the University of Eindhoven in the Netherlands, Scott Ladenheim from Temple University, and Manda Winlaw from Cornell University (the latter three tied for second place). All four speakers gave excellent presentations. It should be stressed that this is the only plenary session at Copper Mountain, all other talks being organized in parallel sessions. This is done in order to give all conference participants the possibility to enjoy the winning student presentations. Michele Benzi (Emory University) and Ray Tuminaro (Sandia National Laboratories) served as conference co-chairs, while Ira Livshits chaired the Student Paper Award Committee. Management services were provided by Annette Anthony of Front Range Scientific Computations, Inc. The Conference gratefully acknowledges financial support from the US Department of Energy, the US National Science Foundation, IBM, Argonne National Laboratory, Lawrence Livermore National Laboratory, Los Alamos National Laboratory, Sandia National Laboratories, and Emory University. The Copper Mountain Conference is organized in cooperation with the Society for Industrial and Applied Mathematics (SIAM) and with the Association for Women in Mathematics (AWM). Submissions to this special section were open to the scientific community, as advertised in advance on web sites of the SIAM Journal on Scientific Computing (SISC) and the Copper Mountain Conference. The 28 papers in this section serve to illustrate the vitality and continuing importance of the ongoing research on iterative methods for a wide variety of areas of applied mathematics and computational science. The guest editorial board worked very hard to ensure a rigorous peer-review process while meeting deadlines. We owe special thanks to Mitch Chernoff (SIAM Publications Manager) and Brittni Holland (SIAM Editorial Associate) for their efforts on this special section. Guest Associate Editors: Xiao-Chuan Cai, Iain Duff, Howard Elman, Kirk Jordan, Tim Kelley, Misha Kilmer, Dana Knoll, Sven Leyffer, Ira Livshits, Tom Manteuffel, Steve McCormick, Luke Olson, John Shadid, David Silvester, Ray Tuminaro, Homer Walker, Carol Woodward, Irad Yavneh

Stochastic Analysis of Project Issues and Fixing its Cause by Project Team and Funding System using Matrix Analytic Method

wo models in this paper with infinite and finite recording of project issues that come up during implementation are treated. In the models when project team and funding are available, issues come up and they are settled once the cause is fixed. In one model infinite number of issues are recorded. In another model the recording is stopped at a level and it is resumed when the recorded number falls due to settlement. Stationary probabilities of number of issues to be fixed and mean are obtained without the use of iterative methods. Matrix analytic method is used and the matrix G has explicit structure. With the same set of all parameters and by varying the issue rate numerical results of both models for comparison are presented.