Convergence Rate Of Iteration Research Articles

General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings

The purpose of this paper is to introduce a new general composite implicit iteration scheme for approximating the common fixed points of a finite family of nonexpansive mappings { T i } i = 1 N in Banach spaces and to prove some weak and strong convergence theorems, this new iteration scheme is defined as follows: x n = α n x n - 1 + ( 1 - α n ) T n y n , y n = r n x n + s n x n - 1 + t n T n x n + w n T n x n - 1 , r n + s n + t n + w n = 1 , { α n } , { r n } , { s n } , { t n } , { w n } ∈ [ 0 , 1 ] , where T n = T n mod N . The general composite implicit iteration scheme presented in this paper included various concrete iteration schemes. Hence, the results presented in this paper extend, generalize and improve the results of Mann, Ishikawa, Xu and Ori, and other authors. Meanwhile, this paper establish the convergence rate optimal model and give the convergence theorems with the estimation of convergence rate.

Algebraic multigrid preconditioner for homogeneous scatterers in electromagnetics

Electromagnetic wave scattering from large and complex bodies is currently the most challenging problem in computational electromagnetics. There is an increasing need for more efficient algorithms with reduced computational complexity and memory requirements. In this work we solve the problem of electromagnetic wave scattering involving three-dimensional, homogeneous, arbitrarily shaped dielectric objects. The fast multipole method (FMM) is used along with the algebraic multigrid (AMG) method, that is employed as a preconditioner, in order to accelerate the convergence rate of the Krylov iterations. Our experimental results suggest much faster convergence compared to the non preconditioned FMM, and hence significant reduction to the overall computation time.

Numerical modeling of crystal growth under strong magnetic fields: An application to the travelling heater method

The article presents a 3-D numerical simulation study for the growth of single crystal semiconductors under strong magnetic fields. In such high fields, the magnetic body force components, which also depend on the flow velocity components present numerical challenges particularly in terms of convergence of iterations. To remedy such difficulties, a novel numerical approach was introduced in an in-house 3-D finite volume-based computer code. As an application, the Travelling Heater Method (THM) was selected for the growth of CdTe crystals under a static vertical magnetic field. The governing equations of the model, which also include the electric charge balance equation in terms of the induced electric potential and the applied magnetic field intensity, are solved numerically. Evolution of the growth interface is simulated with the help of a moving grid algorithm in a block-structured finite-volume code. The convergence rate of iterations is significantly improved by treating a part of the magnetic body force terms implicitly in the iteration loop. Consequently, the magnetic field as high as 15.0 kG did not lead to any convergence problems. Simulation results are presented for the flow, concentration, temperature, and electric potential fields in the liquid solution. Results show that, in spite of the initially assumed-axisymmetric boundary conditions, three-dimensional transport structures develop as soon as the growth process begins. Application of the magnetic field suppresses convection in the solution, and the magnitude of the maximum flow velocity decreases monotonically with the increasing magnetic field intensity.

A generalized approach to formulate the consistent tangent stiffness in plasticity with application to the GLD porous material model

It has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global Newton iterations in the solution of the incremental problem. In this paper, we present a general method to formulate the consistent tangent stiffness for plasticity. The robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with J 2 flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program ABAQUS. The proposed approach is then applied to an anisotropic porous plasticity model, the Gologanu–Leblond–Devaux model. Performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall Newton iterations caused by using the consistent tangent matrix.

Sparse Inverse Preconditioning of Multilevel Fast Multipole Algorithm for Hybrid Integral Equations in Electromagnetics

In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time.

Design and implementation of DIRK integrators for stiff systems

This paper presents new four-stage diagonally implicit Runge–Kutta integration formulas for stiff initial value problems, designed to be L-stable and have optimal order of accuracy. The design makes estimation of local error and interpolation of the solution possible without additional stages. It also enables improved prediction of stage values in a software implementation. Correctly monitoring the convergence rate of simplified Newton iterations improves the implementation too, by providing a sound basis for deciding on Jacobian updates.

A rapid optimization algorithm for GPS data assimilation

Global Positioning System (GPS) meteorology data variational assimilation can be reduced to the problem of a large-scale unconstrained optimization. Because the dimension of this problem is too large, most optimal algorithms cannot be performed. In order to make GPS/MET data assimilation able to satisfy the demand of numerical weather prediction, finding an algorithm with a great convergence rate of iteration will be the most important thing. A new method is presented that dynamically combines the limited memory BFGS (L-BFGS) method with the Hessian-free Newton(HFN) method, and it has a good rate of convergence in iteration. The numerical tests indicate that the computational efficiency of the method is better than the L-BFGS and HFN methods.

Iterative Solution Methods for Modeling Multiphase Flow in Porous Media Fully Implicitly

We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the system into a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.

Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

The convergence rate of the discrete velocity method (DVM), which has been applied extensively in the area of rarefied gas dynamics, is studied via a Fourier stability analysis. The spectral radius of the continuum form of the iteration map is found to be equal to one, which justifies the slow convergence rate of the method. Next the efficiency of the DVM is improved by introducing various acceleration schemes. The new synthetic-type schemes speed up significantly the iterative convergence rate. The spectral radius of the acceleration schemes is also studied and the so-called H1 acceleration method is found to be the optimum one. Finally, the two-dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative DVM. The number of required iterations and the overall computational time are significantly reduced, providing experimental evidence of the analytic formulation. The whole approach is demonstrated using the BGK and S kinetic models.

On the numerical solving of one minimization problem

In this paper we consider the dependence of iterative methods convergence rate on iterative parameters selection. We propose the numerical algorithm determination of the optimal iterative parameters. Using it we solve classical minimization 2D and 3D problems and modificate 2D and 3D minimization problems and analyse the convergence rate dependence on the matrix spectral characteristics.

The positive solution of classical gelfand model with coefficient that change sign

The existence and iteration of positive solution for classical Gelfand models are considered, where the coefficient of nonlinear term is allowed to change sign in [0,1]. By using the monotone iterative technique, an existence theorem of positive solution is obtained, corresponding iterative process and convergence rate are given. This iterative process starts off with zero function, hence the process is simple, feasible and effective.

A domain decomposition method for modelling Stokes flow in porous materials

AbstractAn algorithm is presented for solving the Stokes equation in large disordered two‐dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off‐block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000‐particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd.

A Corotational Formulation for the Simulation of Flexible Mechanisms

A corotational formulation is developed for the extension ofinfinitesimal stiffness and mass matrices to large rotations. Theresulting formulation is applied and certified for a variety of testcases. One aspect which distinguishes the formulation is that not onlythe corotational residual was developed, but the consistent tangentmatrix as well. This fact accelerates the convergence rate of thenonlinear Newmark iterations and guarantees the convergence of the linesearch algorithm. The consistent tangent matrix also opens thepossibility for using the formulation for the study of structuralstability as well.

The analysis of multigrid algorithms for cell centered finite difference methods

In this paper, we examine multigrid algorithms for cell centered finite difference approximations of second order elliptic boundary value problems. The cell centered application gives rise to one of the simplest non-variational multigrid algorithms. We shall provide an analysis which guarantees that the W-cycle and variable V-cycle multigrid algorithms converge with a rate of iterative convergence which can be bounded independently of the number of multilevel spaces. In contrast, the natural variational multigrid algorithm converges much more slowly.

Implicit Lower-Upper/Approximate-Factorization Schemes for Incompressible Flows

A lower-upper/approximate-factorization (LU/AF) scheme is developed for the incompressible Euler or Navier–Stokes equations. The LU/AF scheme contains an iteration parameter that can be adjusted to improve iterative convergence rate. The LU/AF scheme is to be used in conjunction with linearized implicit approximations and artificial compressibility to compute steady solutions, and within sub-iterations to compute unsteady solutions. Formulations based on time linearization with and without sub-iteration and on Newton linearization are developed using spatial difference operators. The spatial approximation used includes upwind differencing based on Roe's approximate Riemann solver and van Leer's MUSCL scheme, with numerically computed implicit flux linearizations. Simple one-dimensional diffusion and advection/diffusion problems are first studied analytically to provide insight for development of the Navier–Stokes algorithm. The optimal values of both time step and LU/AF parameter are determined for a test problem consisting of two-dimensional flow past a NACA 0012 airfoil, with a highly stretched grid. The optimal parameter provides a consistent improvement in convergence rate for four test cases having different grids and Reynolds numbers and, also, for an inviscid case. The scheme can be easily extended to three dimensions and adapted for compressible flows.

A Fast MRQI-Based Algorithm to Locate Minimum Eigenpairs

A method for locating the mlntmum eigenvalue and its corresponding eigenvector is considered. The convergence rate of the Rayleigh quotient iteration (RQI) is cubic. However, unfortunately, the RQI may not always locate the minimum eigenvalue. In this paper, a modified Rayleigh quotient iteration (MRQI) that can always locate the minimum eigenpair is proposed and Mathematical properties of the MRQI are investigated. Based on the MRQI, a fast algorithm to locate minimum eigenpair will be proposed. It can locate multiple minimum eigenvalues, while others which employ an inclusion interval cannot.

Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow

AbstractPeyret (J. Fluid Mech., 78, 49–63 (1976)) and others have described artificial compressibility iteration schemes for solving implicit time discretizations of the unsteady incompressible Navier‐Stokes equations. Such schemes solve the implicit equations by introduing derivatives with respect to a pseudo‐time variable τ and marching out to a steady state in τ. The pseudo‐time evolution equation for the pressure p takes the form ∂p/∂ = −a2∂∇.u, where a is an artificial compressibility parameter and u is the fluid velocity vector. We present a new scheme of this type in which convergence is accelerated by a new procedure for setting a and by introducing an artificial bulk viscosity b into the momentum equation. This scheme is used to solve the non‐linear equations resulting from a fully implicit time differencing scheme for unsteady incompressible flow. We find that the best values of a and b are generally quite different from those in the analogous scheme for steady flow (J. D. Ramshaw and V. A. Mousseau, Comput. Fluids, 18, 361–367 (1990)), owing to the previously unrecognized fact that the character of the system is profoundly altered by the pressence of the physical time derivative terms. In particular, a Fourier dispersion analysis shows that a no longer has the significance of a wave speed for finite values of the physical time step δt,. Inded, if on sets a ˜ |u| as usual, the artificial sound waves cease to exist when δt is small and this adversely affects the iteration convergence rate. Approximate analytical expressions for a and b are proposed and the benefits of their use relative to the conventional values a ∼ |u| and b = 0 are illustrated in simple test calculations.

The Analysis of Multigrid Algorithms for Pseudodifferential Operators of Order Minus One

Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one. Classical multigrid theory deals with the case of differential operators of positive order. The pseudodifferential operator gives rise to a coercive form on ${H^{ - 1/2}}(\Omega )$. Effective multigrid algorithms are developed for this problem. These algorithms are novel in that they use the inner product on ${H^{ - 1}}(\Omega )$ as a base inner product for the multigrid development. We show that the resulting rate of iterative convergence can, at worst, depend linearly on the number of levels in these novel multigrid algorithms. In addition, it is shown that the convergence rate is independent of the number of levels (and unknowns) in the case of a pseudodifferential operator defined by a single-layer potential. Finally, the results of numerical experiments illustrating the theory are presented.

The analysis of multigrid algorithms for pseudodifferential operators of order minus one

Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one. Classical multigrid theory deals with the case of differential operators of positive order. The pseudodifferential operator gives rise to a coercive form on ${H^{ - 1/2}}(\Omega )$. Effective multigrid algorithms are developed for this problem. These algorithms are novel in that they use the inner product on ${H^{ - 1}}(\Omega )$ as a base inner product for the multigrid development. We show that the resulting rate of iterative convergence can, at worst, depend linearly on the number of levels in these novel multigrid algorithms. In addition, it is shown that the convergence rate is independent of the number of levels (and unknowns) in the case of a pseudodifferential operator defined by a single-layer potential. Finally, the results of numerical experiments illustrating the theory are presented.

The iterative solution of Taylor—Galerkin augmented mass matrix equations

AbstractThis paper investigates the convergence properties of iterative schemes for the solution of finite element mass matrix equations that arise through the application of a Taylor‐Galerkin algorithm to solve instationary Navier‐Stokes equations. This is a time‐stepping algorithm that involves Galerkin mass matrix equations at fractional stages within each time‐step. Plane Poiseuille flow and shear‐driven cavity flow are selected as benchmark problems on which to investigate the effects of various choice of scheme and time‐step dependency. The iterative convergence of each mass matrix equation for a single fractional stage is studied, both at the element and the system matrix level. The underlying theory is confirmed and it is shown how optimal iterative convergence rates may be achieved for a Jacobi scheme by employing an appropriate acceleration factor. Moreover, this factor is trivial to compute. The consequential effects on the convergence of the time‐stepping procedure to reach steady‐state are also considered where non‐linear effects are present.