Journal articles: 'Iteration methods'

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Wang, Jin-Ping, Jian-Fei Zhang, Zhi-Guo Qu, and Wen-Quan Tao. "An adaptive inner iterative pressure-based algorithm for steady and unsteady incompressible flows." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 4 (April 15, 2019): 2003–24. http://dx.doi.org/10.1108/hff-09-2018-0483.

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Purpose Pressure-based methods have been demonstrated to be powerful for solving many practical problems in engineering. In many pressure-based methods, inner iterative processes are proposed to get efficient solutions. However, the number of inner iterations is set empirically and kept fixed during the whole computation for different problems, which is overestimated in some computations but underestimated in other computations. This paper aims to develop an algorithm with adaptive inner iteration processes for steady and unsteady incompressible flows. Design/methodology/approach In this work, with the use of two different criteria in two inner iterative processes, a mechanism is proposed to control inner iteration processes to make the number of inner iterations vary during computing according to different problems. By doing so, adaptive inner iteration processes can be achieved. Findings The adaptive inner iterative algorithm is verified to be valid by solving classic steady and unsteady incompressible problems. Results show that the adaptive inner iteration algorithm works more efficient than the fixed inner iteration one. Originality/value The algorithm with adaptive inner iteration processes is first proposed in this paper. As the mechanism for controlling inner iteration processes is based on physical meaning and the feature of iterative calculations, it can be used in any methods where there exist inner iteration processes. It is not limited for incompressible flows. The performance of the adaptive inner iteration processes in compressible flows is conducted in a further study.

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de Lima, Camila, and Elias Salomão Helou. "Fast projection/backprojection and incremental methods applied to synchrotron light tomographic reconstruction." Journal of Synchrotron Radiation 25, no. 1 (January 1, 2018): 248–56. http://dx.doi.org/10.1107/s1600577517015715.

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Iterative methods for tomographic image reconstruction have the computational cost of each iteration dominated by the computation of the (back)projection operator, which take roughlyO(N3) floating point operations (flops) forN×Npixels images. Furthermore, classical iterative algorithms may take too many iterations in order to achieve acceptable images, thereby making the use of these techniques unpractical for high-resolution images. Techniques have been developed in the literature in order to reduce the computational cost of the (back)projection operator toO(N2logN) flops. Also, incremental algorithms have been devised that reduce by an order of magnitude the number of iterations required to achieve acceptable images. The present paper introduces an incremental algorithm with a cost ofO(N2logN) flops per iteration and applies it to the reconstruction of very large tomographic images obtained from synchrotron light illuminated data.

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Rehman, Habib ur, Poom Kumam, Ioannis K. Argyros, Nasser Aedh Alreshidi, Wiyada Kumam, and Wachirapong Jirakitpuwapat. "A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems." Symmetry 12, no. 4 (April 2, 2020): 523. http://dx.doi.org/10.3390/sym12040523.

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The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on a cost bifunction. The method is evaluating only one value of the bifunction per iteration and it is uses an explicit formula for identifying the appropriate stepsize parameter for each iteration. The variable stepsize is going to be effective for enhancing iterative algorithm performance. The variable stepsize is updating for each iteration based on the previous iterations. After numerical examples, we conclude that the effect of the inertial term and variable stepsize has a significant improvement over the processing time and number of iterations.

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Papadrakakis, M. "Accelerating Vector Iteration Methods." Journal of Applied Mechanics 53, no. 2 (June 1, 1986): 291–97. http://dx.doi.org/10.1115/1.3171754.

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This paper describes a technique for accelerating the convergence properties of iterative methods for the solution of large sparse symmetric linear systems that arise from the application of finite element method. The technique is called partial preconditioning process (PPR) and can be combined with pure vector iteration methods, such as the conjugate gradient, the dynamic relaxation, and the Chebyshev semi-iterative methods. The proposed triangular splitting preconditioner combines Evans’ SSOR preconditioner with a drop-off tolerance criterion. The (PPR) is attractive in a FE framework because it is simple and can be implemented at the element level as opposed to incomplete Cholesky preconditioners, which require a sparse assembly. The method, despite its simplicity, is shown to be more efficient on a set of test problems for certain values of the drop-off tolerance parameter than the partial elimination method.

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Proinov, Petko D., and Maria T. Vasileva. "A New Family of High-Order Ehrlich-Type Iterative Methods." Mathematics 9, no. 16 (August 5, 2021): 1855. http://dx.doi.org/10.3390/math9161855.

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One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

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Al-Mohssen, Husain A., Nicolas G. Hadjiconstantinou, and Ioannis G. Kevrekidis. "Acceleration Methods for Coarse-Grained Numerical Solution of the Boltzmann Equation." Journal of Fluids Engineering 129, no. 7 (December 4, 2006): 908–12. http://dx.doi.org/10.1115/1.2742725.

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We present a coarse-grained steady-state solution framework for the Boltzmann kinetic equation based on a Newton-Broyden iteration. This approach is an extension of the equation-free framework proposed by Kevrekidis and coworkers, whose objective is the use of fine-scale simulation tools to directly extract coarse-grained, macroscopic information. Our current objective is the development of efficient simulation tools for modeling complex micro- and nanoscale flows. The iterative method proposed and used here consists of a short Boltzmann transient evolution step and a Newton-Broyden contraction mapping step based on the Boltzmann solution; the latter step only solves for the macroscopic field of interest (e.g., flow velocity). The predicted macroscopic field is then used as an initial condition for the Boltzmann solver for the next iteration. We have validated this approach for isothermal, one-dimensional flows in the low Knudsen number regime. We find that the Newton-Broyden iteration converges in O(10) iterations, starting from arbitrary guess solutions and a Navier-Stokes based initial Jacobian. This results in computational savings compared to time-explicit integration to steady states when the time to steady state is longer than O(40) mean collision times.

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Mat Ali, Nur Afza, Jumat Sulaiman, Azali Saudi, and Nor Syahida Mohamad. "Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation." Indonesian Journal of Electrical Engineering and Computer Science 23, no. 1 (July 1, 2021): 471. http://dx.doi.org/10.11591/ijeecs.v23.i1.pp471-478.

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In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

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Zhou, Ting, and Shi Guang Zhang. "Comparison Results between Jacobi and USSOR Iterative Methods." Advanced Materials Research 989-994 (July 2014): 1790–93. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.1790.

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In this paper, some comparison results between Jacobi and USSOR iteration for solving nonsingular linear systems are presented. It is showed that spectral radius of Jacobi iteration matrix B is less than that of USSOR iterative matrix under some conditions. A numerical example is also given to illustrate our results.

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Gdawiec, Krzysztof, Wiesław Kotarski, and Agnieszka Lisowska. "Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods." Abstract and Applied Analysis 2015 (2015): 1–19. http://dx.doi.org/10.1155/2015/797594.

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A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

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Pour, H. Noormohammadi, and H. Sadeghi Goughery. "Generalized Accelerated Hermitian and Skew-Hermitian Splitting Methods for Saddle-Point Problems." Numerical Mathematics: Theory, Methods and Applications 10, no. 1 (February 2017): 167–85. http://dx.doi.org/10.4208/nmtma.2017.m1524.

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AbstractWe generalize the accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems. These methods involve four iteration parameters whose special choices can recover the preconditioned HSS and accelerated HSS iteration methods. Also a new efficient case is introduced and we theoretically prove that this new method converges to the unique solution of the saddle-point problem. Numerical experiments are used to further examine the effectiveness and robustness of iterations.

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Bai, Zhong-Zhi. "Regularized HSS iteration methods for stabilized saddle-point problems." IMA Journal of Numerical Analysis 39, no. 4 (July 31, 2018): 1888–923. http://dx.doi.org/10.1093/imanum/dry046.

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Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

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Proinov, Petko D. "Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros." Symmetry 13, no. 3 (February 25, 2021): 371. http://dx.doi.org/10.3390/sym13030371.

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In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.

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Lin, R. F., H. M. Ren, Z. Šmarda, Q. B. Wu, Y. Khan, and J. L. Hu. "New Families of Third-Order Iterative Methods for Finding Multiple Roots." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/812072.

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Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.

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Touil, Imene, and Wided Chikouche. "Primal-dual interior point methods for Semidefinite programming based on a new type of kernel functions." Filomat 34, no. 12 (2020): 3957–69. http://dx.doi.org/10.2298/fil2012957t.

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In this paper, we propose the first hyperbolic-logarithmic kernel function for Semidefinite programming problems. By simple analysis tools, several properties of this kernel function are used to compute the total number of iterations. We show that the worst-case iteration complexity of our algorithm for large-update methods improves the obtained iteration bounds based on hyperbolic [24] as well as classic kernel functions. For small-update methods, we derive the best known iteration bound.

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Daengsaen, Jukkrit, and Anchalee Khemphet. "On the Rate of Convergence of P-Iteration, SP-Iteration, and D-Iteration Methods for Continuous Nondecreasing Functions on Closed Intervals." Abstract and Applied Analysis 2018 (July 2, 2018): 1–6. http://dx.doi.org/10.1155/2018/7345401.

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We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.

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Axelsson, Owe. "Milestones in the Development of Iterative Solution Methods." Journal of Electrical and Computer Engineering 2010 (2010): 1–33. http://dx.doi.org/10.1155/2010/972794.

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Iterative solution methods to solve linear systems of equations were originally formulated as basic iteration methods of defect-correction type, commonly referred to as Richardson's iteration method. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR) method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iteration methods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.

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Ivannikov, A. D., and A. L. Stempkovskiy. "Iterative Methods for Solving Systems of Multi-Valued Logical Equations in the Simulation of Object Control Digital Systems." Mekhatronika, Avtomatizatsiya, Upravlenie 21, no. 9 (September 7, 2020): 511–20. http://dx.doi.org/10.17587/mau.21.511-520.

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The article is devoted to the analysis of methods for solving systems of multivalued logical equations by iteration methods. Iterative methods for solving such systems of equations are a mathematical description of the main process of functional-logical simulation, which is used at the stage of designing digital systems for objects control to verify the correctness of the design. Consideration of multi-valued values of logical signals at the outputs of blocks and elements of digital systems is explained by the fact that in some cases, to analyze the correctness of time relationships when simulating the hardware of digital systems, a several valued representation of logical signals is used, as well as that recently, logical elements are being developed that implement four or more valued logic. Based on the analysis of the structure of the system of logical equations used in digital hardware simulation, using graph and logical models, an analysis is made of the existence of solutions and their number. Iterative methods of a simple and generalized iteration are analyzed, a relationship is shown between the number of solutions of the system of equations and its graph representation, which reflects a given circuit of connecting elements of the hardware of a digital control system. For the generalized iteration method, options with a different structure of the iteration trace are considered, in particular, it is shown that, with a certain structure of the iteration trace, the generalized iteration turns into a simple iteration or Seidel iteration. It is shown that the generalized iteration most adequately describes the process of simulating the switching of logical signals in a simulated circuit of digital control systems hardware. The correspondence between various options of functional-logical simulation of digital systems and the used methods of iterative solution of systems of logical equations is shown.

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Walker, W. Thomas, Thomas F. Rossi, and Nazrul Islam. "Method of Successive Averages Versus Evans Algorithm: Iterating a Regional Travel Simulation Model to the User Equilibrium Solution." Transportation Research Record: Journal of the Transportation Research Board 1645, no. 1 (January 1998): 32–40. http://dx.doi.org/10.3141/1645-05.

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The results of comparative tests of two methods for iterating a regional travel demand model system are presented. Model iteration is necessary to ensure consistency between model input and output speeds, as required by current federal legislation. Two methods were tested: the Evans algorithm and the method of successive averages. A series of tests using alternative assignment techniques was conducted for each method. Criteria for evaluating the iteration methods included convergence error, average highway speeds compared with observations, highway vehicle miles traveled compared with Highway Performance Monitoring System estimates, transit boardings compared with observations, and computer running time. It was concluded that the Evans algorithm performed the best, primarily on the basis of superior computational efficiency, although good results were obtained by using the method of successive averages. Use of the Evans algorithm is recommended, embedded within a formal assignment restart, for iterating the model system. Multiple iterations of highway assignment should be used in the initial model loop and all-or-nothing assignments in subsequent iterations of the modeling chain.

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Uc-Cetina, Víctor, Francisco Moo-Mena, and Rafael Hernandez-Ucan. "Composition of Web Services Using Markov Decision Processes and Dynamic Programming." Scientific World Journal 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/545308.

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We propose a Markov decision process model for solving the Web service composition (WSC) problem. Iterative policy evaluation, value iteration, and policy iteration algorithms are used to experimentally validate our approach, with artificial and real data. The experimental results show the reliability of the model and the methods employed, with policy iteration being the best one in terms of the minimum number of iterations needed to estimate an optimal policy, with the highest Quality of Service attributes. Our experimental work shows how the solution of a WSC problem involving a set of 100,000 individual Web services and where a valid composition requiring the selection of 1,000 services from the available set can be computed in the worst case in less than 200 seconds, using an Intel Core i5 computer with 6 GB RAM. Moreover, a real WSC problem involving only 7 individual Web services requires less than 0.08 seconds, using the same computational power. Finally, a comparison with two popular reinforcement learning algorithms, sarsa andQ-learning, shows that these algorithms require one or two orders of magnitude and more time than policy iteration, iterative policy evaluation, and value iteration to handle WSC problems of the same complexity.

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Wang, Zhuande, Chuansheng Yang, and Yubo Yuan. "Convergence Results on Iteration Algorithms to Linear Systems." Scientific World Journal 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/273873.

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In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of the Jacobi iterative matrix is positive and the one of backward iterative matrix is strongly positive (lager than a positive constant). Secondly, the mentioned two iterations have the same convergence results (convergence or divergence simultaneously). Finally, some numerical experiments show that the proposed algorithms are correct and have the merit of backward methods.

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Zeng, Min-Li, and Guo-Feng Zhang. "A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems." East Asian Journal on Applied Mathematics 6, no. 4 (October 19, 2016): 367–83. http://dx.doi.org/10.4208/eajam.150116.240516a.

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AbstractIn this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSS-based iteration methods are proposed for solving weakly nonlinear systems based on separable property of the linear and nonlinear terms. The conditions for guaranteeing the local convergence are studied and the quasi-optimal iterative parameters are derived. Numerical experiments are implemented to show that the new methods are feasible and effective for large scale systems of weakly nonlinear systems.

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Zhang, Yanmei, Xia Cui, and Guangwei Yuan. "Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (June 26, 2020): 1465–90. http://dx.doi.org/10.1051/m2an/2019095.

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This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.

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BIAN, WENSHENG, and BILL POIRIER. "ACCURATE AND HIGHLY EFFICIENT CALCULATION OF THE O(1D)HCl VIBRATIONAL BOUND STATES, USING A COMBINATION OF METHODS." Journal of Theoretical and Computational Chemistry 02, no. 04 (December 2003): 583–97. http://dx.doi.org/10.1142/s0219633603000768.

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Hypochlorous acid, HOCl, is an important intermediate in the O (1D) HCl reactive system. Due in part to a large number of vibrational bound states (over 800), extremely large direct product basis sets (around 300,000) are required to compute the energy levels just below the dissociation threshold. This situation, combined with a very high density of states, results in difficult convergence for iterative methods — e.g. Lanczos requires 50,000 iterations, and filter diagonalization uses 60,000 iterations. In contrast, using new methodologies, we are able to compute the highest-lying bound states with only 271 iterations, although the CPU cost per iteration is substantially greater. Lower lying states are also computed, for a fraction of the CPU cost of the highest energy calculation.

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Da Costa, Felipe Orlando, Felipe Leonardo Barcelos Mateus, and Irineu Petri Júnior. "Influence of partitioning methods on computational cost of cfd simulations applied to hydrocyclones." Journal of Engineering and Exact Sciences 6, no. 4 (October 21, 2020): 0528–32. http://dx.doi.org/10.18540/jcecvl6iss4pp0528-0532.

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In simulations, when using multipartitioned computers, the way in which the mesh is partitioned directly affects the average time per iteration, and therefore, the computational cost. This study proposes an analysis of the average time per iteration of 23 different partition methods available for tridimensional mesh in software FLUENT 19.2. For the calculation of the average iteration time, 100 iterations were used. Generally, the best partitioning methods were those in which the mesh division was made perpendicularly to the axis of the equipment. It was stated the choice of an adequate partitioning method can save high costs of computational power. For the hydrocyclone studied, with a computer with 8 cores, approximately 24.56 hours of simulation were saved, representing almost 20% of the total time.

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Liu, Zhongyun, Xiaorong Qin, Nianci Wu, and Yulin Zhang. "The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices." Canadian Mathematical Bulletin 60, no. 4 (December 1, 2017): 807–15. http://dx.doi.org/10.4153/cmb-2016-077-5.

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AbstractIt is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., T = C−S with C a circulantmatrix and S a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.

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Liu, Yi Di. "Research on Iterative Method in Solving Linear Equations on the Hadoop Platform." Applied Mechanics and Materials 347-350 (August 2013): 2763–68. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.2763.

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Solving linear equations is ubiquitous in many engineering problems, and iterative method is an efficient way to solve this question. In this paper, we propose a general iteration method for solving linear equations. Our general iteration method doesnt contain denominators in its iterative formula, and this relaxes the limits that traditional iteration methods require the coefficient aii to be non-zero. Moreover, as there is no division operation, this method is more efficient. We implement this method on the Hadoop platform, and compare it with the Jacobi iteration, the Guass-Seidel iteration and the SOR iteration. Experiments show that our proposed general iteration method is not only more efficient, but also has a good scalability.

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Barakat, Saman A., and Qusay I. Sarhan. "Performance evaluation of list iteration methods in Java: an empirical study." Innovaciencia Facultad de Ciencias Exactas, Físicas y Naturales 6, no. 1 (December 28, 2018): 1–6. http://dx.doi.org/10.15649/2346075x.467.

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Introduction: Lists are used in various software applications including web applications, desktop applications, and Internet of Things (IoT) applications to store different types of items (e.g. country name, product model, and device category). Users can select one or more of these items to perform specific tasks such as filling forms, ordering products, reading device data, etc. In some software applications, lists store a huge number of items to be iterated over in order to know what users have selected. From a software development perspective, there are a number of methods to iterate over list items. Materials and Methods: In this paper, five list iteration methods: Classic For, Enhanced For, Iterator, List Iterator, and For Each have been compared experimentally with each other with regard to their performance (execution time required to iterate over list items). Thus, a number of experimental test scenarios have been conducted to obtain comparable results. Results and Discussion: The experimental results of this study have been presented in Table 4. Conclusions: Overall performance evaluation showed that Iterator and List Iterator methods outperformed other list iteration methods in all test scenarios. However, List Iterator outperformed Iterator when the list size was small. On the other hand, Iterator outperformed List Iterator when the list size was large.

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Li, Yazhe, Kai Zhou, and Zhen Zhang. "The flow-difference feedback iteration method for aerostatic thrust bearings and its convergence characteristics." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 233, no. 11 (April 29, 2019): 1743–52. http://dx.doi.org/10.1177/1350650119846230.

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The flow-difference feedback iteration method for aerostatic thrust bearings is proposed to the flow balance-based iteration, and two modification methods are further provided to improve the adaptability. The bearing capability calculated by the proposed method is validated by the experimental data. Moreover, the influence of convergence rate factors, iterative initial values, and mesh grids on the iteration ratio is investigated. Compared with the conventional iteration methods, the proposed method with appropriate convergence rate factors provides a higher convergence efficiency. In addition, good convergence behavior under different iterative initial values and the mesh grid size is shown, and the convergence rate is insensitive to the finite difference method parameters. A series of calculations are conducted to investigate the generality of the proposed methods.

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Tian, Zhaolu, Xiaoyan Liu, Yudong Wang, and P. H. Wen. "The modified matrix splitting iteration method for computing PageRank problem." Filomat 33, no. 3 (2019): 725–40. http://dx.doi.org/10.2298/fil1903725t.

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In this paper, based on the iteration methods [3,10], we propose a modified multi-step power-inner-outer (MMPIO) iteration method for solving the PageRank problem. In the MMPIO iteration method, we use the multi-step matrix splitting iterations instead of the power method, and combine with the inner-outer iteration [24]. The convergence of the MMPIO iteration method is analyzed in detail, and some comparison results are also given. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithm.

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Cao, Jing. "Inner Sequential Single Solid Method for Layout Optimization of Multi-Materials." Journal of Physics: Conference Series 2235, no. 1 (May 1, 2022): 012091. http://dx.doi.org/10.1088/1742-6596/2235/1/012091.

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Abstract In multiple materials layout optimization, the optimal solution is difficult to achieve due to the number of design variables being too large and the material interpolation scheme becoming complicated when using density-like method. To bypass this problem, an inner sequential single solid optimization (ISSSO) method is presented in this study. There are two types of iterations in the optimization process of this method: the inner and outer iterations. In the inner iteration, the original layout optimization of many materials is replaced with a series of single solid sub-optimizations. In each sub-optimization, only one solid is chosen to update and the rest keeps unchanged. The inner iteration stops when all the solids are updated consequently. The next loop of inner iteration starts when the current solution does not meet the requirement of convergence. The outer iteration starts only if the current inner iteration comes to an end. In the outer iteration, the new inner iterations for update of all the solid materials are carried out. The effectiveness of ISSSO method is verified by comparing the calculation results with other methods.

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Chen, Yong-Lin, and Xue-Yuan Tan. "Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices." Applied Mathematics and Computation 167, no. 2 (August 2005): 930–56. http://dx.doi.org/10.1016/j.amc.2004.06.143.

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Zeng, Min-Li, and Guo-Feng Zhang. "On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations." Mathematics 8, no. 2 (February 6, 2020): 208. http://dx.doi.org/10.3390/math8020208.

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To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods.

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Gazzola, Silvia, Sebastian James Scott, and Alastair Spence. "Flexible Krylov Methods for Edge Enhancement in Imaging." Journal of Imaging 7, no. 10 (October 18, 2021): 216. http://dx.doi.org/10.3390/jimaging7100216.

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Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner–outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner–outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub–Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.

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Nawaz, Muhammad, Amir Naseem, and Waqas Nazeer. "New iterative methods using variational iteration technique and their dynamical behavior." Open Journal of Mathematical Analysis 2(2018), no. 2 (December 31, 2018): 1–9. http://dx.doi.org/10.30538/psrp-oma2018.0013.

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Lubyshev, Fedor V., and Mahmut E. Fairuzov. "On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 21, no. 3 (September 30, 2019): 329–42. http://dx.doi.org/10.15507/2079-6900.21.201903.329-342.

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An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.

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Ali, M. S. S. "Descent methods for convex optimization problems in Banach spaces." International Journal of Mathematics and Mathematical Sciences 2005, no. 15 (2005): 2347–57. http://dx.doi.org/10.1155/ijmms.2005.2347.

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We consider optimization problems in Banach spaces, whose cost functions are convex and smooth, but do not possess strengthened convexity properties. We propose a general class of iterative methods, which are based on combining descent and regularization approaches and provide strong convergence of iteration sequences to a solution of the initial problem.

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Mungkasi, Sudi. "Successive Approximation, Variational Iteration, and Multistage-Analytical Methods for a SEIR Model of Infectious Disease Involving Vaccination Strategy." Communication in Biomathematical Sciences 3, no. 2 (May 10, 2021): 114–26. http://dx.doi.org/10.5614/cbms.2020.3.2.3.

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We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

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Wang, Shou Jun, and Ming Wei Wei. "Application of Accelerated Iterative Method in Calculating Wave Length in Harbor Engineering." Applied Mechanics and Materials 130-134 (October 2011): 3481–84. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3481.

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In this paper, a specific application accelerated iterative method is presented for calculating wave length in harbor engineering, which includes calculation method of wave length and specific implement in Excel. Different wavelengths into the iteration formula to calculate the same result can be obtained, but the calculation speed of different methods have significant differences to arrive at the fastest method . Calculated by accelerating the iteration method can significantly increase the computing speed and calculation steps. After the derivation of several methods and calculations show that Newton iteration is the fastest way to convergence speed, in the practical range of about 10 steps through the iterative convergence results can be obtained.

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Mrkaic, Mico. "Policy iteration accelerated with Krylov methods." Journal of Economic Dynamics and Control 26, no. 4 (April 2002): 517–45. http://dx.doi.org/10.1016/s0165-1889(00)00073-7.

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Maasoey, Svein‐Erik, Bjoern Angelsen, and Trond Varslot. "Iteration of ultrasound aberration correction methods." Journal of the Acoustical Society of America 115, no. 5 (May 2004): 2522. http://dx.doi.org/10.1121/1.4783289.

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Deng, Youjun, and Zhenhai Liu. "Iteration methods on sideways parabolic equations." Inverse Problems 25, no. 9 (August 14, 2009): 095004. http://dx.doi.org/10.1088/0266-5611/25/9/095004.

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Arnold, M. "Constraint partitioning in dynamic iteration methods." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 81, S3 (2001): 735–38. http://dx.doi.org/10.1002/zamm.200108115143.

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Abdon Atangana. "A NEW DEFY FOR ITERATION METHODS." Journal of Applied Analysis & Computation 5, no. 3 (2015): 273–83. http://dx.doi.org/10.11948/2015025.

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Bru, Rafael, Ludwig Elsner, and Michael Neumann. "Models of parallel chaotic iteration methods." Linear Algebra and its Applications 103 (May 1988): 175–92. http://dx.doi.org/10.1016/0024-3795(88)90227-3.

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Hong, Baojian, and Dianchen Lu. "Modified Fractional Variational Iteration Method for Solving the Generalized Time-Space Fractional Schrödinger Equation." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/964643.

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Based on He’s variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations.

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Hemeda, A. A., and E. E. Eladdad. "New Iterative Methods for Solving Fokker-Planck Equation." Mathematical Problems in Engineering 2018 (November 28, 2018): 1–9. http://dx.doi.org/10.1155/2018/6462174.

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In this article, we propose the new iterative method and introduce the integral iterative method to solve linear and nonlinear Fokker-Planck equations and some similar equations. The results obtained by the two methods are compared with those obtained by both Adomian decomposition and variational iteration methods. Comparison shows that the two methods are more effective and convenient to use and overcome the difficulties arising in calculating Adomian polynomials and Lagrange multipliers, which means that the considered methods can simply and successfully be applied to a large class of problems.

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Xu, Kai, and Zhi Xiong. "Nonparametric Tensor Completion Based on Gradient Descent and Nonconvex Penalty." Symmetry 11, no. 12 (December 12, 2019): 1512. http://dx.doi.org/10.3390/sym11121512.

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Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization problem and designs an efficient iterative method to solve it. In each iteration, we not only calculate the missing entries by the aid of data correlation, but consider the low-rank of tensor and the convergence speed of iteration. Our iteration is based on the gradient descent method, and approximates the gradient descent direction with tensor matricization and singular value decomposition. Considering the symmetry of every dimension of a tensor, the optimal unfolding direction in each iteration may be different. So we select the optimal unfolding direction by scaled latent nuclear norm in each iteration. Moreover, we design formula for the iteration step-size based on the nonconvex penalty. During the iterative process, we store the tensor in sparsity and adopt the power method to compute the maximum singular value quickly. The experiments of image inpainting and link prediction show that our method is competitive with six state-of-the-art methods.

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Nasikun, Ahmad, and Klaus Hildebrandt. "The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems." ACM Transactions on Graphics 41, no. 2 (April 30, 2022): 1–14. http://dx.doi.org/10.1145/3495208.

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Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM) , a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.

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Lotfi, Taher, and Tahereh Eftekhari. "A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations." Chinese Journal of Mathematics 2014 (March 13, 2014): 1–7. http://dx.doi.org/10.1155/2014/369713.

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Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

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Li, Meng, and Yong Zhang. "Convergence Optimization and Verification for Single-Channel Remote Parameter Control of a Nonlinear System." Applied Sciences 9, no. 3 (February 6, 2019): 549. http://dx.doi.org/10.3390/app9030549.

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In this paper, the theory of RPC (remote parameter control) iteration process of linear situation without and with iteration coefficient as well as nonlinear situation with coefficient is analyzed. The influence of iteration coefficient on iterative convergence control condition is analyzed. Two kinds of optimized control method for iteration coefficient based on the system transfer function estimation are proposed. A lightweight motorcycle and electro-hydraulic servo road vibration test bench are used to verify the feasibility of the optimized control methods for the reproduction of road profiles of SISO (single-input, single-output) system. According to the experiment result, which is the RMS (root mean square) of the NSRE (normalized spectrum of response error) of system, the convergent precision, convergent speed and iteration stability are discussed to present the advantage and disadvantage of the optimized control methods. Compared with three commonly used manual methods, the result shows the rapidity and stability of optimized control methods.

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Journal articles on the topic 'Iteration methods'

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