Convergence Rate Of Iterative Methods Research Articles

Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

Additional Results on Convergence of Alternating Iterations Involving Rectangular Matrices

ABSTRACTDifferent classes of matrix splittings of semi-monotone matrices lead to many comparison results, which are useful tools in examining the convergence rate of iterative methods for solving rectangular systems of linear equations in a faster way. In this context, the theory of alternating iterations for rectangular matrices was introduced recently, in order to arrive at the desired solution of accuracy or at the exact solution. In this article, we expand convergence theory of such alternating iterations and obtain comparison results for such iterations.

Hybrid Accelerated Method via Volume-Surface Integral Equation for Efficient Calculation of Monostatic RCS

A novel efficient hybrid accelerated method is proposed for the fast analysis of the monostatic electromagnetic scattering problems arising from volume-surface integral equations (VSIE) formulation. In the first step, by utilizing the low rank property, several largest eigenvalues and corresponding eigenvectors of the multiple right hand sides can be computed and saved efficiently by adaptive cross approximation (ACA) algorithm. The iterative solution of linear equations is required at these principle eigenvectors. Compared with solving linear equations at each angle repeatedly, the proposed method is able to greatly reduce the number of equations. In the second step, a disturbed symmetric successive over-relaxation (D-SSOR) preconditioner is constructed to speed up the convergence rate of iterative methods. Numerical results demonstrate that the present method can reduce the computational time significantly for monostatic VSIE calculation with high accuracy.

Convergence Results of the Improving Modified Gauss-Seidel (IMGS) Iterative Method for a Symmetric Positive Definite Matrix

In this paper, in order to improve the convergence rates of iterative method solving the linear system, the improving modified Gauss-Seidel (IMGS) iterative method with a preconditioner is proposed. Some convergence and comparison results are given when is a symmetric definite matrix are provided.

Rapid springback compensation for age forming based on quasi Newton method

Iterative methods based on finite element simulation are effective approaches to design mold shape to compensate springback in sheet metal forming. However, convergence rate of iterative methods is difficult to improve greatly. To increase the springback compensate speed of designing age forming mold, process of calculating springback for a certain mold with finite element method is analyzed. Springback compensation is abstracted as finding a solution for a set of nonlinear functions and a springback compensation algorithm is presented on the basis of quasi Newton method. The accuracy of algorithm is verified by developing an ABAQUS secondary development program with MATLAB. Three rectangular integrated panels of dimensions 710 mm ×750 mm integrated panels with intersected ribs of 10 mm are selected to perform case studies. The algorithm is used to compute mold contours for the panels with cylinder, sphere and saddle contours respectively and it takes 57%, 22% and 33% iterations as compared to that of displacement adjustment (DA) method. At the end of iterations, maximum deviations on the three panels are 0.618 4 mm, 0.624 1 mm and 0.342 0 mm that are smaller than the deviations determined by DA method (0.740 8 mm, 0.740 8 mm and 0.713 7 mm respectively). In following experimental verification, mold contour for another integrated panel with 400 mm×380 mm size is designed by the algorithm. Then the panel is age formed in an autoclave and measured by a three dimensional digital measurement devise. Deviation between measuring results and the panel’s design contour is less than 1 mm. Finally, the iterations with different mesh sizes (40 mm, 35 mm, 30 mm, 25 mm, 20 mm) in finite element models are compared and found no considerable difference. Another possible compensation method, Broyden-Fletcher-Shanmo method, is also presented based on the solving nonlinear functions idea. The Broyden-Fletcher-Shanmo method is employed to compute mold contour for the second panel. It only takes 50% iterations compared to that of DA. The proposed method can serve a faster mold contour compensation method for sheet metal forming.

Application of Hierarchical Two-Level Spectral Preconditioning Method for Electromagnetic Scattering from the Rough Surface

The higher-order hierarchical Legendre basis functions combining the electrical field integral equations (EFIE) are developed to solve the scattering problems from the rough surface. The hierarchical two-level spectral preconditioning method is developed for the generalized minimal residual iterative method (GMRES). The hierarchical two-level spectral preconditioner is constructed by combining the spectral preconditioner and sparse approximate inverse (SAI) preconditioner to speed up the convergence rate of iterative methods. The multilevel fast multipole method (MLFMM) is employed to reduce memory requirement and computational complexity of the method of moments (MoM) solution. The accuracy and efficiency are confirmed with a couple of numerical examples.

Convergence of the Preconditioned AOR Method for an H-Matrix

Linear system with H-matrix often appears in a wide variety of areas and is studied by many numerical researchers. In order to improve the convergence rates of iterative method solving the linear system whose coefficient matrix is an H-matrix. In this paper, a preconditioned AOR iterative method with a multi-parameters preconditioner with a general upper triangular matrix is proposed. In addition, the convergence of the coressponding iterative method are established. Lastly, we provide numerical experiments to illustrate the theoretical results.

The modified multilevel compressed block decomposition algorithms for analyzing the scattering of objects in half space

The convergence rate of iterative methods can vary in an unpredictable way. It is related to the matrix condition number, which is notoriously bad for the electric field integral equation in the large-scale electromagnetic problems. Therefore, an efficient direct solution—a multilevel compressed block decomposition (MLCBD) algorithm based on the adaptive cross-approximation algorithm—is applied to overcome this problem; it is very efficient for the monostatic problems. Simulation results of the objects up and below ground in half space demonstrate that the proposed MLCBD method is efficient for analyzing electromagnetic problems. Copyright © 2013 John Wiley & Sons, Ltd.

BLOCK MATRIX PRECONDITIONER METHOD FOR THE ELECTRIC FIELD INTEGRAL EQUATION (EFIE) FORMULATION BASED ON LOOP-STAR BASIS FUNCTIONS

In this paper, electromagnetic scattering problems are analyzed using an electric fleld integral equation (EFIE) formulation that is based on loop-star basis functions so as to avoid low-frequency instability problems. Moreover, to improve the convergence rate of iterative methods, a block matrix preconditioner (BMP) is applied to the EFIE formulation based on loop star-basis functions. Because the matrix system resulting from the conventional method of moments is a dense matrix, a sparse matrix version of each block matrix is constructed, followed by the inversion of the resultant block sparse matrix using incomplete factorization. Numerical results show that the proposed BMP is e-cient in terms of computation time and memory usage.

A Modified SSOR Preconditioning Strategy for Helmholtz Equations

The finite difference method discretization of Helmholtz equations usually leads to the large spare linear systems. Since the coefficient matrix is frequently indefinite, it is difficult to solve iteratively. In this paper, a modified symmetric successive overrelaxation (MSSOR) preconditioning strategy is constructed based on the coefficient matrix and employed to speed up the convergence rate of iterative methods. The idea is to increase the values of diagonal elements of the coefficient matrix to obtain better preconditioners for the original linear systems. Compared with SSOR preconditioner, MSSOR preconditioner has no additional computational cost to improve the convergence rate of iterative methods. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners.

Expository Research Papers

The two papers in this issue deal with differential equations, one with the numerical solution of partial differential equations, and the other one with analytic solutions for ordinary differential equations. In his paper From Functional Analysis to Iterative Methods, Robert Kirby is concerned with linear systems arising from discretizations of partial differential equations (PDEs). Specifically, the PDEs are elliptic and describe boundary value problems; the discretizations are done via finite elements, and at issue is the convergence rate of iterative methods for solving the linear systems. The author's approach is to go back to the underlying variational problem in a Hilbert space, and to make ample use of the Riesz representation theorem. This point of view results in short and elegant proofs, as well as the construction of efficient preconditioners. The general theory is illustrated with two concrete model problems of PDEs for convection diffusion and planar elasticity. This paper will appeal to anybody who has an interest in the numerical solution of PDEs. In 1963 the mathematician/meteorologist Edward Lorenz formulated a system of three coupled nonlinear ordinary differential equations, whose long-term behavior is described by an attractor with fractal structure. You can see a beautiful rendition of the thus named Lorenz attractor on the cover of this issue. Although it is easy to plot solutions of the Lorenz system, it is much harder to determine them mathematically. This is what motivated the paper Complex Singularities and the Lorenz Attractor by Divakar Viswanath and Sonmez Sahutoglu. Their idea is to allow the time variable to be complex, rather than real; to focus on singular solutions; and to express these singular solutions in terms of so-called psi series. After all is said and done, the authors end up with a two-parameter family of complex solutions to the Lorenz system. This a highly readable and very enjoyable paper, with concrete steps for future research, and connections to thunderstorms and analytic function theory.

Shifted SSOR preconditioning technique for electromagnetic wave scattering problems

AbstractTo efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix‐vector product operations. The symmetric successive over‐relaxation (SSOR) preconditioner is constructed based on the near‐field matrix of the EFIE and employed to speed up the convergence rate of iterative methods. This technique can be greatly improved by shifting the near‐field matrix of the EFIE with the principle value term of the magnetic field integral equation (MFIE) operator. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1035–1039, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24254

APPLICATION OF TWO-STEP SPECTRAL PRECONDITIONING TECHNIQUE FOR ELECTROMAGNETIC SCATTERING IN A HALF SPACE

To e-ciently solve large dense complex linear system arising from electric fleld integral equations (EFIE) formulation of half-space electromagnetic scattering problems, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix- vector product operations. The two-step spectral preconditioning is developed for the generalized minimal residual iterative method (GMRES). The two-step spectral preconditioner is constructed by combining the spectral preconditioner and sparse approximate inverse (SAI) preconditioner to speed up the convergence rate of iterative methods. Numerical experiments for scattering from conducting objects above or embeded in a lossy half-space are given to demonstrate the e-ciency of the proposed method.

SSOR PRECONDITIONED INNER-OUTER FLEXIBLE GMRES METHOD FOR MLFMM ANALYSIS OF SCATTERING OF OPEN OBJECTS

To efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix-vector product operations. The inner-outer flexible generalized minimum residual method (FGMRES) is combined with the symmetric successive over- relaxation (SSOR) preconditioner based on the near-part matrix of the EFIE in the inner iteration of FGMRES to speed up the convergence rate of iterative methods. Numerical experiments with a few electromagnetic scattering problems for open structures are given to demonstrate the efficiency of the proposed method.

A multi-parameters preconditioned AOR iterative method for linear systems

In order to improve the convergence rates of iterative method solving the linear system Ax = b , a preconditioned AOR iterative method with a multi-parameters preconditioner I + S ∼ α is proposed. Some convergence and comparison results for α i ∈ [ 0 , 1 ] are given when A is an L-matrix. When A is an H-matrix, the convergence of the corresponding methods is discussed for α i ∈ [ 0 , α i ′ ) , where α i > 1 . Furthermore, effectiveness of the new preconditioner method is shown by numerical experiments. Numerical experiments show that our methods are superior to the basic AOR iterative methods.

The use of preconditioning in direct methods of solving positive definite linear systems

The concept of preconditioning hitherto successfully applied to accelerate the convergence rate of iterative methods is now extended to the direct methods of solution of small order dense illconditioned matrix systems. It is shown that by solving the equivalent preconditioned linear system using Gaussian elimination and Choleski factorisation methods minimises the rounding errors incurred in the computational process. Theoretical analysis justifying this theory are included together with supporting numerical evidence.

Diagonal preconditioners for the EFIE using a wavelet basis

The electric field integral equation (EFIE) has found widespread use and in practice has been accepted as a stable method. However, mathematically, the solution of the EFIE is an ill-posed problem. In practical terms, as one uses more and more expansion and testing functions per wavelength, the condition number of the resulting moment-method matrix increases (without bound). This means that for high-sampling densities, iterative methods such as conjugate gradients converge more slowly. However, there is a way to change all this. The EFIE is considered using a wavelet basis for expansion and for testing functions. Then, the resulting matrix is multiplied on both sides by a diagonal matrix. This results in a well-conditioned matrix which behaves much like the matrix for the magnetic field integral equation (MFIE). Consequences for the stability and convergence rate of iterative methods are described.

Simplified heuristic Fourier analysis of iteration convergence rates

AbstractChan and Elman (1989) have recently shown that heuristic Fourier analysis can be used to determine approximate convergence rates of iterative methods. Here we present a simple procedure and useful relations that facilitate the practical application of this method. The procedure is based on the fact that the mesh spacing h is typically small, so that attention may be restricted to the asymptotic behaviour of the convergence rate R for small h. We accordingly derive simple expressions for the first few coefficients in a Taylor series expansion of R in powers of h. These coefficients are expressed in terms of derivatives with respect to h of the complex Fourier growth factor ξ rather than its modulus |ξ|. The required derivatives can then be obtained by implicit differentiation without explicitly solving for ξ, which further simplifies the analysis. The overall procedure is illustrated by applying it to the Jacobi, Gauss‐Seidel and successive overrelaxation methods, for which we obtain results asymptotically equivalent to those of Chan and Elman. We also analyse the effect of second‐order Richardson acceleration on the Jacobi method, for which we obtain an asymptotic convergence rate in precise agreement with classical analysis.

On accelerating the convergence rate of iterative methods in elastic–plastic stress analysis

On accelerating the convergence rate of iterative methods in elastic–plastic stress analysis

On the Convergence of Some Generalized Preconditioned Iterative Methods

This paper generalizes the preconditioning techniques introduced by Evans [2] for the numerical solution of the linear system $Au = b$ and defines two extrapolated versions of the Gauss–Siedel method as well as an extrapolated version of the successive overrelaxation method. Finally, it establishes some convergence theorems for the considered iterative schemes when A has particular properties such as being consistently ordered, positive definite having weak diagonal dominance, etc...