CG-type Method Research Articles

Weighted Conjugate Gradient-Type Methods for Solving Quadrature Discretization of Fredholm Integral Equations of the First Kind

A variant of conjugate gradient-type methods, called weighted conjugate gradient (WCG), is given to solve quadrature discretization of various first-kind Fredholm integral equations with continuous kernels. The WCG-type methods use a new inner product instead of the Euclidean one arising from discretization of $$L^2$$ -inner product by the quadrature formula. On this basis, the proposed algorithms generate a sequence of vectors which are approximations of solution at the quadrature points. Numerical experiments on a few model problems are used to illustrate the performance of the new methods compared to the CG-type methods.

On preconditioned BiCGSTAB solver for MLPG method applied to heat conduction in 3D complex geometry

Development of meshfree methods is in full swing in the last few decades as these methods overcome the shortcomings of mesh based methods. Development of meshfree methods for large scale problems requires fast iterative solvers. Preconditioner plays an important role in improving convergence rate of an iterative solver. Recently proposed SRJ method can be used as a stand-alone solver and as a preconditioner in CG type methods. Objective of the present work is to evaluate the performance of SRJ method as a preconditioner in BiCGSTAB method in connection with the MLPG method. Three dimensional Poisson equation and heat conduction equation are solved in regular and irregular domain respectively. The SRJ method is tested as a solver and as a preconditioner in BiCGSTAB method. Its performance as a preconditioner is compared with Jacobi and SOR preconditioners.

辺有限要素法による電磁界解析におけるブロックマルチカラーオーダリングを援用した並列化前処理付きMRTR法に基づく線形方程式求解の高速化

To realize fast electromagnetic field analysis, the parallelization technique has been often introduced into the preconditioned Krylov subspace method. When the multicolor ordering is applied to parallelization of forward and backward substitution, the elapsed time of matrix calculation might increase owing to the increment of bandwidth. Therefore, the block-multicolor ordering based on the level structure arising in reverse Cuthill-McKee ordering has been developed. The validity of developed method was demonstrated on the parallelized incomplete-Cholesky-preconditioned conjugate gradient (ICCG) method. In this paper, the parallelization performance of preconditioned minimized residual method based on the three-term recurrence formula of the CG-type (MRTR) method supported by developed ordering is investigated. Furthermore, the affinity of developed ordering or cache-cache elements technique for parallelized forward and backward substitution in Eisenstat's technique is particularly examined.

Parallelization of Preconditioned MRTR Method With Eisenstat’s Technique by Means of Algebraic Multicolor Ordering

The performance of preconditioned Minimized Residual method based on the Three-term Recurrence formula of the conjugate gradient CG-type (MRTR) method has been demonstrated on various symmetric linear systems based on 3-D FEM. The symmetric Gauss–Seidel-preconditioned MRTR method with Eisenstat’s technique (MESGS-MRTR) has higher validity for reducing the elapsed time when a conventional PC is used. However, MESGS-MRTR cannot be parallelized, when a block preconditioner is adopted. Therefore, we propose a parallelized MESGS-MRTR method supported by algebraic multicolor ordering in the 3-D low-frequency electromagnetic problems. We present a performance comparison of the proposed method with block preconditioner using reverse Cuthill–McKee ordering.

Monotonicity of error of regularized solution and its use for parameter choice

We consider an ill-posed equation in a Hilbert space with a noisy operator and a noisy right-hand side. The noise level information is given in a general form, as a norm of a certain operator applied to the noise. We derive the monotone error rule (ME-rule) for the choice of the regularization parameter in many methods, giving parameter such that the error is monotonically increasing for larger parameters in the Tikhonov method and for smaller stopping indices in iteration methods. Regularization methods considered include -scale regularization in (iterated) Tikhonov method and in iteration methods (Landweber method, CG type methods, semi-iterative methods). We also consider modifications of the ME-rule and show in numerical experiments (test problems from Hansen’s Regularization Toolbox, including the sideways heat equation) their advantages over the discrepancy principle.

Improvement of the Preconditioned MRTR Method With Eisenstat's Technique in Real Symmetric Sparse Matrices

The Incomplete Cholesky Conjugate Gradient (ICCG) method is widely used to solve indefinite algebraic equations obtained using an edge-based finite element method. However, when a linear solver based on the minimum residual is used, there is a possibility of reducing the elapsed time for a linear system. This paper shows the performance of the preconditioned minimized residual method based on the three-term recurrence formula of the CG-type (MRTR) method by comparing the MRTR method with the ICCG method for real symmetric sparse matrices. Furthermore, we intend to reduce computational costs by using Eisenstat's technique, and achieve more speed-up by applying a preconditioned residual to the convergence criterion.

A CG-Type Method for Inverse Quadratic Eigenvalue Problems in Model Updating of Structural Dynamics

AbstractIn this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matricesM, DandKfor the quadratic pencilQ(λ) =λ2M+ λD+K, so thatQ(λ) has a prescribed subset of eigenvalues and eigenvectors. This method can determine the solvability of the inverse eigenvalue problem automatically. We then consider the least squares model for updating a quadratic pencilQ(λ). More precisely, we update the model coefficient matrices M, C and K so that (i) the updated model reproduces the measured data, (ii) the symmetry of the original model is preserved, and (iii) the difference between the analytical triplet (M, D, K) and the updated triplet (Mnew,Dnew,Knew) is minimized. In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.

Global SCD algorithm for real positive definite linear systems with multiple right-hand sides

In the present paper, we present the left conjugate direction (LCD) method for linear systems with multiple right-hand sides. We define the method by using left and right conjugate direction matrices and Petrov–Galerkin condition. The method has no breakdown for real positive definite systems. The method reduces to the usual CG-type method when A is symmetric positive definite. We also show how to apply these methods for solving the Lyapunov matrix equation. Finally a numerical experiment is given to illustrate the result.

Approximation of Scattered Data by Trigonometric Polynomials on the Torus and the 2-sphere

Fast, efficient and reliable algorithms for the discrete least-square approximation of scattered points on the torus Td and the sphere S2 by trigonometric polynomials are presented. The algorithms are based on iterative CG-type methods in combination with fast Fourier transforms for nonequispaced data. The emphasis is on numerical aspects, in order to solve large scale problems. Numerical examples show the efficiency of the new algorithms.

레이레이 계수의 최소화에 의한 내부고유치 계산을 위한 병렬준비행렬들의 비교

Recently, CG (Conjugate Gradient) scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for interior eigenvalues for the following eigenvalue problem, Ax＝λx (1) The given matrix A is assummed to be large and sparse, and symmetric. Also, the method is very amenable to parallel computations. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. We compare the parallel preconditioners for the computation of the interior eigenvalues of a symmetric matrix by CG-type method. The considered preconditioners are Point-SSOR, ILU (0) in the multi-coloring order, and Multi-Color Block SSOR (Symmetric Succesive OverRelaxation). We conducted our experiments on the CRAY­T3E with 128 nodes. The MPI (Message Passing Interface) library was adopted for the interprocessor communications. The test matrices are up to 512×512 in dimensions and were created from the discretizations of the elliptic PDE. All things considered the MC-BSSOR seems to be most robust preconditioner.

Accuracy of preconditioned CG-type methods for least squares problems

The conjugate gradient method with IMGS, an incomplete modified version of Gram-Schmidt orthogonalization to obtain an incomplete orthogonal factorization preconditioner, applied to the normal equations (PCGLS) is often used as the basic iterative method to solve the linear least squares problems. In this paper, a detailed analysis is given for understanding the effect of rounding errors on IMGS and determining the accuracy of computed solutions of PCGLS with IMGS for linear least squares problems in finite precision. It is shown that for a consistent system, the difference between the true residuals and the updated approximate residual vectors generated depends on the machine precision ε, on the maximum growth in norm of the iterates over their initial values, the norm of the true solution, and the condition number of R which is affected by the drop set in incomplete Gram-Schmidt factorization. Similar results are obtained for the difference between the true and computed solution for inconsistent systems. Numerical tests are carried out to confirm the theoretical conclusions.

Numerical experiments to optimize the use of (I)LU preconditioning in the iterative linear solver package LINSOL

In the iterative linear solver package LINSOL several generalized conjugate gradient (CG) methods (or, briefly, CG-type methods) with quite different properties are implemented. With these methods polyalgorithms with automatic method switching are constructed. The “emergency exit” that is taken in the worst case is the ATPRES method (which is very robust, but very slow). In this paper we investigate if (I)LU preconditioning were a better emergency exit and how the drop tolerance for small elements in ILU affects the convergence behavior. The answer will be: it depends.

An engineering approach to generalized conjugate gradient methods and beyond

This is an elementary introductory paper (rather a tutorial) for generalized conjugate gradient (CG) methods. The emphasis is on “understanding” the approach of the now rather sophisticated CG-type methods and beyond. Practical aspects like normalization and smoothing are discussed. Then the development of different types of methods with their characteristic properties is presented. For a model of the Navier-Stokes equations that varies between strong and weak ellipticity, the methods are compared and their typical properties are demonstrated.

Indefinite elliptic problem preconditioning

Based on investigating the eigenvalue distribution of the preconditioned matrices that arise in the finite-element discretization of indefinite elliptic problems with preconditioning matrices, constructed on the basis of the principal coercive part of the considered elliptic problems, it is proved that the Krylov subspace CG-type methods like MINRES converge with a rate of convergence determined only by the stability properties of the discrete problem in terms of a priori estimates and the accuracy of the preconditioning matrix with respect to the principal coercive part of the considered finite-element stiffness matrix. This is also demonstrated by numerical experiments. The method is of practical interest if there are only a few negative eigenvalues.

Solving general sparse linear systems using conjugate gradient-type methods

The problem of finding an approximation of @@@@ = A † b (where A † is the pseudo-inverse of A ∈ @@@@ m @@@@ n with m ≥ n and rank ( A ) = n ) is discussed. It is assumed that A is sparse but has neither a special pattern (as bandedness) nor a special property (as symmetry or positive definiteness). In this paper it is shown that preconditioners obtained by neglecting small elements during the decomposition of A into easily invertible matrices can be used efficiently with conjugate gradient-type methods if an adaptive strategy for deciding when an element is small is implemented. The resulting preconditioned methods are often better than the corresponding direct and pure iterative methods or those based on preconditionings in which elements are neglected when they appear in a predefined set of positions in the matrix, i.e. positional rather than numerical dropping. Numerical results are given to illustrate the performance of the CG-type methods preconditioned via numerical dropping.

SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems

SIRT and CG-type methods have been successfully employed for the approximate solution of least-squares problems arising in tomography. In this paper we study and compare their convergence and regularization properties. It is pointed out that SIRT methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system, whereas CG-type methods do not. For a large class of model problems it is shown that virtually the same solutions as obtained by SIRT methods can be obtained by applying a CG-type method to a properly rescaled system, but with an amount of work proportional to the square root of the amount of work with SIRT.

Preconditioned CG-type methods for solving the coupled system of fundamental semiconductor equations

This paper presents some of the authors' experimental results in applying Preconditioned CG-type methods to nonsymmetric systems of linear equations arising in the numerical solution of the coupled system of fundamental stationary semiconductor equations. For this type of problem it is shown that these iterative methods are efficient both in computation times and in storage requirements. All results have been obtained on an HP 350 computer.