Chebyshev Semi-iterative Method Research Articles

An improved Chebyshev semi-iterative method for simulating rainfall infiltration in unsaturated soils and its application to shallow landslides

A Chebyshev semi-iterative method (CSIM) with preconditioner (P-CSIM) is proposed to simulate numerically unsteady flow in unsaturated soils. An exponential function was used to linearize Richards’ equation, which was discretized using a finite difference method to obtain the linear equations. For layered soils, the numerical discretization takes into account the continuity of the pressure head at the interface. Conventional iterative approaches, including the Jacobi and the Gauss–Seidel iterative methods, require more iterations to obtain numerical solutions. The CSIM incurs more computational cost per iteration. The improved P-CSIM by comparison solves the linear equations in fewer iterations, with higher computational efficiency, and better robustness. The results indicate that the proposed P-CSIM provides good computational acceleration; specifically, at least 50% more than the CSIM and the Gauss–Seidel and Jacobi iterative methods. To simulate shallow landslides, the stability analysis of infinite slopes was developed to incorporate the time-varying pressure head and soil–water characteristic curves of unsaturated soils. The results of the examples indicate that the solver is able also to represent the increase of pressure head which may lead to a landslide. As a result, the proposed method with high computational efficiency can be developed to effectively analyze rainfall-induced shallow landslides in unsaturated soils.

An Acceleration Method for Stationary Iterative Solution to Linear System of Equations

An acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided. AMS subject classifications: 65F10, 15A06

LSRN: A PARALLEL ITERATIVE SOLVER FOR STRONGLY OVER- OR UNDERDETERMINED SYSTEMS.

We describe a parallel iterative least squares solver named LSRN that is based on random normal projection. LSRN computes the min-length solution to min x∈ℝ n ‖Ax - b‖2, where A ∈ ℝ m × n with m ≫ n or m ≪ n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and LSRN automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size ⌈γ min(m, n)⌉ × min(m, n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results show that on a shared-memory machine, LSRN is very competitive with LAPACK's DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in. Further experiments show that LSRN scales well on an Amazon Elastic Compute Cloud cluster.

Quasi-Chebyshev accelerated iteration methods based on optimization for linear systems

In this paper, we present a quasi-Chebyshev accelerated iteration method for solving a system of linear equations. Compared with the Chebyshev semi-iterative method, the main difference is that the parameter ω is not obtained by a Chebyshev polynomial but by optimization models. We prove that the quasi-Chebyshev accelerated iteration method is unconditionally convergent if the original iteration method is convergent, and also discuss the convergence rate. Finally, three numerical examples indicate that our method is more efficient than the Chebyshev semi-iterative method.

An Acceleration Method for Stationary Iterative Solution to Linear System of Equations

AbstractAn acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided.

The convergence rate of the Chebyshev semiiterative method under a perturbation of the foci of an elliptic domain

The Chebyshev semiiterative method (CHSIM) is a powerful method for finding the iterative solution of a nonsymmetric real linear system Ax = b if an ellipse excluding the origin well fits the spectrum of A. The asymptotic rate of convergence of the CHSIM for solving the above system under a perturbation of the foci of the optimal ellipse is studied. Several formulae to approximate the asymptotic rates of convergence, up to the first order of a perturbation, are derived. These generalize the results about the sensitivity of the asymptotic rate of convergence to a perturbation of a real-line segment spectrum by Hageman and Young, and by the first author. A numerical example is given to illustrate the theoretical results.

Is a Chebyshev method optimal for an elliptic region also optimal for a nearly elliptic region?

The asymptotic rate of convergence of an optimal Chebyshev semiiterative method for solving a real and nonsymmetric linear system x =T x + c can be improved by the related (2,2)-step iterative methods under certain conditions. The condition for which a Chebyshev method asymptotically optimal for an elliptic region is also asymptotically optimal for a nearly elliptic region is presented. Thus a (2,2)-step method is asymptotically superior to the Chebyshev method asymtotically optimal for a nearly elliptic region under certain conditions. A numerical example illustrates our results.

Equivalent iterative methods for p-cyclic matrices

An overview is given of the simplifications which arise when p-cyclic systems are solved by iterative methods. Besides the classic iterative methods, we treat the Chebyshev semi-iterative method and the OR and MR variants of the class of Krylov subspace methods. Particular emphasis is given to equivalent iterations applied to the cyclically reduced system.

Acceleration of the outer iterations of the space-dependent neutron kinetics equations solution

A variant of the Chebyshev iterative procedure, called the Chebyshev Semi-Analytical (CSA) method, is developed for acceleration of outer iterations of the space-dependent neutron kinetics equations solution. A Chebyshev polynomials sequence is chosen to optimize the convergence of residual vectors to the eigenvector of the iteration matrix. When the convergence is attained, the “exact” solution is computed by adding an error vector to the obtained approximate solution. Convergence rate of the CSA method is determined by the second eigenvalue of the iteration matrix in contrast to the Chebyshev Semi-Iterative (CSI) method where it is defined by the spectral radius of the same matrix. An adaptive Chebyshev procedure is modified for estimation of the optimum iteration parameters of the CSA method. The proposed method is implemented in the 3D neutron kinetics computer code SKETCH. The numerical results of benchmark problem calculations demonstrate that the CSA method is 2–5 times faster than the CSI method for delayed critical transient calculations. The relative effectiveness of the CSA method increases with increase in time step size.

Approximating dominant singular triplets of large sparse matrices via modified moments

A procedure for determining a few of the largest singular values and corresponding singular vectors of large sparse matrices is presented. Equivalent eigensystems are solved using a technique originally proposed by Golub and Kent based on the computation of modified moments. The asynchronicity in the computations of moments and eigenvalues makes this method attractive for parallel implementations on a network of workstations. Although no obvious relationship between modified moments and the corresponding eigenvectors is known to exist, a scheme to approximate both eigenvalues and eigenvectors (and subsequently singular values and singular vectors) has been produced. This scheme exploits both modified moments in conjunction with the Chebyshev semi-iterative method and deflation techniques to produce approximate eigenpairs of the equivalent sparse eigensystems. The performance of an ANSI-C implementation of this scheme on a network of UNIX workstations and a 256-processor Cray T3D is presented.

The convergence rate of the Chebyshev sim under a perturbation of a complex line-segment spectrum

The Chebyshev semiiterative method ( chsim) is probably the best known and most often used method for the iterative solution of linear system x = Tx + c, where the spectrum of T is located in a complex line segment [α, β] excluding 1. The asymptotic convergence factor (ACF) of the chsim, under a perturbation of [α, β], is considered. Several formulae for the approximation to the ACFs, up to the second order of a perturbation, are derived. This generalizes the results about the sensitivity of the asymptotic rate of convergence to the estimated eigenvalues by Hageman and Young in the case that both α and β are real. Two numerical examples are given to illustrate the theoretical results.

Applying the JII-SI, and CG methods to two classes of linear problems

A Chebyshev semiiterative (JII-SI) method based on the optimum Jacobi second-degree iterative (JII) method is developed. A comparison of both methods and the conjugate gradient (CG) method is made by applying them to the two following classes of algebraic linear problems: systems with positive definite and consistently ordered matrix, and systems with positive definite and generalized stochastic matrix.

Estimating the largest singular values of large sparse matrices via modified moments

We describe a procedure for determining a few of the largest singular values of a large sparse matrix. The method by Golub and Kent which uses the method of modified moments for estimating the eigenvalues of operators used in iterative methods for the solution of linear systems of equations is appropriately modified in order to generate a sequence of bidiagonal matrices whose singular values approximate those of the original sparse matrix. A simple Lanczos recursion is proposed for determining the corresponding left and right singular vectors. The potential asynchronous computation of the bidiagonal matrices using modified moments with the iterations of an adapted Chebyshev semi-iterative (CSI) method is an attractive feature for parallel computers. Comparisons in efficiency and accuracy with an appropriate Lanczos algorithm (with selective re-orthogonalization) are presented on large sparse (rectangular) matrices arising from applications such as information retrieval and seismic reflection tomography. This procedure is essentially motivated by the theory of moments and Gauss quadrature.

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.

Vectorized and Multitasked Solution of the Few-Group Neutron Diffusion Equations

A numerical algorithm with parallelism was used to solve the two-group, multidimensional neutron diffusion equations on computers characterized by shared memory, vector pipeline, and multi-CPU architecture features. Specifically, solutions were obtained on the Cray X/MP-48, the IBM-3090 with vector facilities, and the FPS-164. The material-centered mesh finite difference method approximation and outer-inner iteration method were employed. Parallelism was introduced in the inner iterations using the cyclic line successive overrelaxation iterative method and solving in parallel across lines. The outer iterations were completed using the Chebyshev semi-iterative method that allows parallelism to be introduced in both space and energy groups. For the three-dimensional model, power, soluble boron, and transient fission product feedbacks were included. Concentrating on the pressurized water reactor (PWR), the thermal-hydraulic calculation of moderator density assumed single-phase flow and a closed flow channel, allowing parallelism to be introduced in the solution across the radial plane. Using a pinwise detail, quarter-core model of a typical PWR in cycle 1, for the two-dimensional model without feedback the measured million floating point operations per second (MFLOPS)/vector speedups were 83/11.7. 18/2.2, and 2.4/5.6 on the Cray, IBM, and FPS without multitasking, respectively. Lower performance was observed with a coarser mesh, i.e., shorter vectormore » length, due to vector pipeline start-up. For an 18 x 18 x 30 (x-y-z) three-dimensional model with feedback of the same core, MFLOPS/vector speedups of --61/6.7 and an execution time of 0.8 CPU seconds on the Cray without multitasking were measured. Finally, using two CPUs and the vector pipelines of the Cray, a multitasking efficiency of 81% was noted for the three-dimensional model.« less

Application of the Chebyshev method to radiative transfer calculations for laser heated targets

The Chebyshev semi-iterative method is used to accelerate the Λ iteration for solving the radiative transfer equation for a spectral line. For conditions typical of laser heated targets this method may be more efficient than those used in present calculations.

ADI as a Preconditioning for Solving the Convection-Diffusion Equation

We examine a splitting of the operator obtained from a steady convection-diffusion equation with variable coefficients in which the convection term dominates. The operator is split into a dominant and a subdominant parts consistent with the inherent directional property of the partial differential equation. The equations involving the dominant parts should be easily solved. We accelerate the convergence of this splitting or the outer iteration by a Chebyshev semi-iterative method. When the dominant part has constant coefficients, it can be easily solved using alternating direction implicit (ADI) methods. This is called the inner iteration. The optimal parameters for a stationary two-parameter ADI method are obtained when the eigenvalues become complex. This corresponds to either the horizontal or the vertical half-grid Reynolds number larger than unity. The Chebyshev semi-iterative method is used to accelerate the convergence of the inner ADI iteration. A two-fold increase in speed is obtained when the ADI iteration matrix has real eigenvalues, and the increase is less significant when the eigenvalues are complex. If either the horizontal or the vertical half-grid Reynolds number is equal to one, the spectral radius of the optimal ADI iterative matrix is zero. However, a high degree of nilpotency impairs rapid convergence. This problem is removed by introducing a more implicit iterative method called ADI/Gauss–Seidel (ADI/GS). ADI/GS resolves the nilpotency and, thus, converges more rapidly for half-grid Reynolds number near 1. Finally, our methods are compared with several well-known schemes on test problems.

Solving Theodorsen's integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods

We investigate several iterative methods for the numerical solution of Theodorsen's integral equation, the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors. All our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast Fourier transform for this. In particular, we discuss the nonlinear JOR iteration, the nonlinear SOR iteration, a nonlinear second order Euler iteration, the nonlinear Chebyshev semi-iterative method, and its cyclic variant. Under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters (e.g., the underrelaxation factor) and the asymptotic convergence factor. Weaker related results for the general non-symmetric case are presented too. Practically, our methods extend the range of application of Theodorsen's method and improve its effectiveness strikingly.

Second order Chebyshev semi-iteration in connection with p-parametric E.A.D.I. schemes

In this paper an analysis of a second order Chebyshev semi-iterative method for the p-parametric E.A.D.I. schemes is presented with a recommendation for the selection of the optimum parameters in order to achieve the maximum rate of convergence.

On the relation between dynamic relaxation and semi-iterative matrix methods

The well known relation between some matrix iterative methods for the solution of elliptic partial differential equations and time dependent parabolic equations is extended to establish a correspondence between the Chebyshev semi-iterative method and a time dependent hyperbolic equation. The method of "dynamic relaxation" based upon finding the transient solution to the corresponding dynamical system of equations is shown to be an exact analogue of the Chebyshev method provided the appropriate choice of inertia and damping coefficients is made. The possibility of developing further techniques based upon different physical models and leading to improved convergence is noted.