Generalized Conjugate Residual Method Research Articles

Structure-preserving Quaternion Generalized Conjugate Residual Method

Structure-preserving Quaternion Generalized Conjugate Residual Method

An Efficient and Error-Controllable Angular Sweep Algorithm for Electromagnetic Wave Scattering and its Analysis

The angular sweep of electromagnetic wave scattering is formulated as a matrix equation with multiple right-hand sides (RHSs). Although the low-rank approximation of an RHS matrix is a popular choice for reducing the computational costs of multiple RHSs, only a small amount of research has been conducted to explore how this approximation impacts the solution quality. Furthermore, there has not been sufficient research on the quality of the solution as a function of the accuracy of the iterative solver. We present an error analysis of the approximated solution considering both the reduced number of RHSs and the tolerance of the iterative solver. Based on the error analysis, a new angular sweep algorithm is proposed with fine-tuned tolerances of the iterative solver for individual singular vectors. The different tolerances for each singular vector increase the efficiency of the proposed algorithm. Another benefit of the proposed algorithm is that the error can be bounded by a user-defined global tolerance. In addition, a variant of the generalized conjugate residual method for multiple RHSs is introduced to accelerate iterative solvers. Finally, numerical validation is conducted with three examples in which the discontinuous Galerkin surface integral equation method is applied. The experiments support two conclusions: tight upper and lower bounds of the solution error exist, and fine-tuning the tolerances reduces the computational costs.

Implementation and performance evaluation of a communication-avoiding GMRES method for stencil-based code on GPU cluster

In this study, a communication-avoiding generalized minimum residual method (CA-GMRES) is implemented on a hybrid CPU–GPU cluster, targeted for the performance acceleration of iterative linear system solver in the gyrokinetic toroidal five-dimensional Eulerian code (GT5D). In the GT5D, its sparse matrix-vector multiplication operation (SpMV) is performed as a 17-point stencil-based computation. The specialized part for the GT5D is only in the SpMV, and the other parts are usable also for other application program codes. In addition to the CA-GMRES, we implement and evaluate a modified variant of CA-GMRES (M-CA-GMRES) proposed in the previous study Idomura et al. (in: Proceedings of the 8th workshop on latest advances in scalable algorithms for large-scale systems (ScalA ’17), 2017. https://doi.org/10.1145/3148226.3148234) to reduce the amount of floating-point calculations. This study demonstrates that beneficial features of the CA-GMRES are in its minimum number of collective communications and its highly efficient calculations based on dense matrix–matrix operations. The performance evaluation is conducted on the Reedbush-L GPU cluster, which contains four NVIDIA Tesla P100 (Pascal GP100) GPUs per compute node. The evaluation results show that the M-CA-GMRES or CA-GMRES for the GT5D is advantageous over the GMRES or the generalized conjugate residual method (GCR) on GPU clusters, especially when the problem size (vector length) is large so that the cost of the SpMV is less dominant. The M-CA-GMRES is 1.09 ×, 1.22 × and 1.50 × faster than the CA-GMRES, GCR and GMRES, respectively, when 64 GPUs are used.

Bucket-based multigrid preconditioner for solving pressure Poisson equation using a particle method

The execution time for incompressible particle methods is dominated by solving large sparse linear systems. In this study, a novel simple bucket-based multigrid (BMG) preconditioner is presented to retrieve linear scaling. In the algorithm, the domain is decomposed into cubic boxes, to enable recursive aggregation of the closest neighboring particles. Under a moving particle semi-implicit (MPS) discretization scheme, parametric, verification, and performance studies were conducted for basic problems. From the parametric study, the BMG preconditioner was accompanied with a Krylov subspace accelerated multigrid cycle strategy and incorporated into a generalized conjugate residual method. Regardless of dimension and degree of particle distribution irregularity, the method significantly outperformed a conjugate gradient (CG), an incomplete Cholesky CG, and a conventional plain aggregation-based algebraic multigrid preconditioned solver. For a dam-break problem with 1.3 M and 2.3 M particles in 2D and 3D, the proposed method, as a linear solver for the MPS method, was 27 and 5.4 times faster than the CG method.

GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization

Purpose This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method. Design/methodology/approach The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗. Findings The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver. Originality/value This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

Subspace recycling accelerates the parametric macro‐modeling of MEMS

SUMMARYA fast computational technique that speeds up the process of parametric macro‐model extraction is proposed. An efficient starting point is the technique of parametric model order reduction (PMOR). The key step in PMOR is the computation of a projection matrix V, which requires the computation of multiple moment matrices of the underlying system. In turn, for each moment matrix, a linear system with multiple right‐hand sides has to be solved. Usually, a considerable number of linear systems must be solved when the system includes more than two free parameters. If the original system is of very large size, the linear solution step is computationally expensive. In this paper, the subspace recycling algorithm outer generalized conjugate residual method combined with generalized minimal residual method with deflated restarting (GCRO‐DR), is considered as a basis to solve the sequence of linear systems. In particular, two more efficient recycling algorithms, G‐DRvar1 and G‐DRvar2, are proposed. Theoretical analysis and simulation results show that both the GCRO‐DR method and its variants G‐DRvar1 and G‐DRvar2 are very efficient when compared with the standard solvers. Furthermore, the presented algorithms overcome the bottleneck of a recently proposed subspace recycling method the modified Krylov recycling generalized minimal residual method. From these subspace recycling algorithms, a PMOR process for macro‐model extraction can be significantly accelerated. Copyright © 2013 John Wiley & Sons, Ltd.

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

This work is concerned with the development and study of a minimum residual norm subspace method based on the generalized conjugate residual method with inner orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of nonsymmetric or non-Hermitian linear systems. First we recall the main features of flexible generalized minimum residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

Scalable robust solvers for unstructured FE geodynamic modeling applications: Solving the Stokes equation for models with large localized viscosity contrasts

The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area. Solving high‐resolution convection problems with order of magnitude 108 degrees of freedom requires solvers that scale well, with respect to both the number of degrees of freedom as well as having optimal parallel scaling characteristics on computer clusters. We investigate the use of a smoothed aggregation (SA) algebraic multigrid (AMG)‐type solution strategy to construct efficient preconditioners for the Stokes equation. We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating the convergence rate of the generalized conjugate residual method (GCR). We abbreviate this procedure as BTA‐GCR. For our experiments, we use unstructured grids with quadratic finite elements, making the model flexible with respect to geometry and topology and O(h3) accurate. We find that AMG‐type methods scale linearly (O(n)), with respect to the number of degrees of freedom, n. Although not all parts of AMG have preferred parallel scaling characteristics, we show that it is possible to tune AMG, resulting in parallel scaling characteristics that we consider optimal, for our experiments with up to 100 million degrees of freedom. Furthermore, AMG‐type methods are shown to be robust methods, allowing us to solve very ill‐conditioned systems resulting from strongly varying material properties over short distances in the model interior.

Efficient Analysis and Design of Finite Phased Arrays of Printed Dipoles Using Fast Algorithm: Some Case Studies

This paper presents an accurate and efficient method of moments (MoM) analysis for characterizing a large-scaled phased dipole array. In this method, the mixed potential integral equation (MPIE) is utilized and solved iteratively using the generalized conjugate residual method. The precorrected fast Fourier transform (pFFT) method is employed to accelerate the entire computational process so as to reduce significantly both the memory requirement and the computational time for designing large arrays. Numerical results of an example, which had to be solved using approximate approach due to the large number of elements, are presented to demonstrate the efficiency and accuracy of the present method.

Analysis and numerical computation of diffraction of an optical field by a subwavelength-size aperture in a thick metallic screen by use of a volume integral equation.

Diffraction of an optical field by an aperture in a thick metallic screen is analyzed numerically by use of a three-dimensional volume integral equation together with a generalized conjugate residual method and fast Fourier transformation. Numerical results were validated by reciprocity and the independence of the results of the truncated discretized volume size used in numerical calculations. Near and far fields of square, circular, and triangular apertures in a thick screen are obtained numerically. Some of the numerical results obtained in the present study agree with previously reported experimental results. The surface plasmon polaritons excited on the sidewalls of the aperture can explain the basic characteristics of near-field distribution of apertures. The Bethe-Bouwkamp theory was found to be insufficient to explain the basic characteristics of the near field around the subwavelength aperture in a practical metallic screen.

A fast analysis of scattering and radiation of large microstrip antenna arrays

An accurate and efficient method that combines the precorrected fast Fourier transform (FFT) method and the discrete complex image method (DCIM) is presented to characterize the scattering and radiation properties of arbitrarily shaped microstrip patch antennas. In this method, the mixed potential integral equation (MPIE) is discretized in the spatial domain by means of the discrete complex image method. The resultant system is solved iteratively using the generalized conjugate residual method (GCR) and the precorrected-FFT technique is used to speed up the matrix-vector multiplication. The precorrected-FFT eliminates the need to generate and store the usual square impedance matrix and thus leads to a significant reduction in memory requirement and computational cost. Numerical results are presented for arbitrarily shaped microstrip antenna arrays to demonstrate the accuracy and efficiency of this technique.

2D CAVITY MODELING USING METHOD OF MOMENTS AND ITERATIVE SOLVERS - Abstract

The method of moments (MoM) and the electric field integral equations (EFIEs), for both parallel and perpendicular polarization were applied to simulate scattering from 2D cavity structures. This code employed several matrix equation solvers, such as the LU decomposition, conjugate gradient (CG) method, bi-conjugate gradient (BCG) method, generalized conjugate residual (GCR) method, and generalized minimal residual (GMRES) method. The simulated results can be used for future reference and benchmarking. A comparison on the convergence behavior of the CG, BCG, GCR, and GMRES methods was made for the benchmark geometry, such as offset bend cavity, rectangular waveguide with hub, double-bend S-shaped cavity, etc. Some comments on the performance of the various iterative solvers will be highlighted.

DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems

In a recent work by the author [Linear Algebra Appl. 298 (1999) 99] Krylov subspace methods were derived for Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b, where A∈ C N×N is a singular and in general non-hermitian matrix that has arbitrary index. One of these methods, modeled after the Generalized Conjugate Residual method (GCR) and denoted DGCR, is considered in the present work again. It is shown that all of the approximations produced by DGCR exist, and a GMRES like algorithm, denoted DGMRES, for its implementation is derived. Like GMRES, DGMRES too is economical computationally and storagewise.

The pgcr method for solving unsymmetric linear systems on a vector multiprocessor

In this paper the algorithm of the preconditioned generalized conjugate residual method for solving unsymmetric linear systems on a vector multiprocessor are proposed, when A is five, seven or nine-diagonal matrix. The convergence of these iterative methods is analysed. We show that for this algorithm the number of iterations is about the same as for the multiprocessor PGCR algorithms. The resulting preconditioning GCR method has been tested on simulation parallel-vector computer. Numerical examples indicate that the new algorithm is very efficient, since the vector multiprocessor computation can be applied.

Application of preconditioned conjugate gradient-like methods to reactor kinetics

Several conjugate gradient-like (CG-like) methods are applied to solve the nonsymmetric linear systems of equations derived from the time-dependent two-dimensional two-energy-group neutron diffusion equations by a finite difference approximation. The CG-like methods are: the generalized conjugate residual method; the generalized conjugate gradient least-square method; the generalized minimal residual method (GMRES); the conjugate gradient square method; and a variant of bi-conjugate gradient method (Bi-CGSTAB). In order to accelerate these CG-like methods, several preconditioning techniques are investigated. Two of the preconditioners are based on pointwise incomplete factorization: the incomplete LU (ILU) and the modified incomplete LU (MILU) decompositions. Another two, based on the block tridiagonal structure of the coefficient matrix, are blockwise and modified blockwise incomplete factorizations, BILU and MBILU. Finally, the last two are the alternating-direction implicit and symmetric successive overrelaxation (SSOR) preconditioners, which are derived from the basic iterative schemes. Comparisons are made by using CG-like methods combined with different preconditioners to solve a sequence of time-step reactor transient problems. We also compare the numerical results of CG-like methods with that of the line SOR method and the hybrid SSOR-SI Krylov subspace method, which is the Chebyshev semi-iteration method with the preconditioner SSOR, and with the GMRES method to estimate the extreme eigenvalues for the acceleration parameters. Numerical tests indicate that preconditioned BI-CGSTAB with the preconditioner MBILU requires less CPU time and number of iterations than other methods in reactor kinetics computation. Another advantage shown is that the preconditioneed CG-like methods are less sensitive to the time-step size used than the Chebyshev semi-iteration method and line SOR method. Comparison between CG-like methods indicates that the CGS, Bi-CGSTAB and GMRES methods are, on average, better than the other methods in reactor kinetics computation. Moreover, numerical results indicate that a good preconditioner is more important than the choice of CG-like methods. Numerical results also show that the MILU decomposition based on the conventional row-sum criterion has difficulty yielding a convergent solution in reactor kinetics computation. An improved version of the MILU decomposition is introduced by using Stone's idea for reactor kinetics computation.

Conjugate residual methods for almost symmetric linear systems

This paper concerns the use of conjugate residual methods for the solution of nonsymmetric linear systems arising in applications to differential equations. We focus on an application derived from a seismic inverse problem. The linear system is a small perturbation to a symmetric positive-definite system, the nonsymmetries arising from discretization errors in the solution of certain boundary-value problems. We state and prove a new error bound for a class of generalized conjugate residual methods; we show that, in some cases, the perturbed symmetric problem can be solved with an error bound similar to the one for the conjugate residual method applied to the symmetric problem. We also discuss several applications for special distributions of eigenvalues.

On vectorizing the preconditioned generalized conjugate residual methods

A numerical study of the efficiency of the vectorized generalized conjugate residual methods (GCR) is performed using three different preconditioners, incomplete LU factorization, diagonal scaling and polynomial. The GCR behaviour is valued in connection with the solution of large, sparse unsymmetric systems of equations, arising from the finite element integration of the diffusion-convection equation. The size of the test problems ranges from 509 to 1700. Results from a set of numerical experiments are presented. The speed-up obtained is up to 11 times over the best scalar implementation. All the experiments were carried out on the vector computer Cray X-MP/48.

Preconditioned iterative algorithms for large sparse unsymmetric problems

AbstractA numerical study of the efficiency of the generalized conjugate residual methods (GCR) is performed using three different preconditioners all based upon an incomplete LU factorization. The GCR behavior is evaluated in connection with the solution of large, sparse unsymmetric systems of equations, arising from the finite element integration of the diffusion‐convection equation for 2‐dimensional (2‐D) and 3‐D problems with different Peclet and Courant numbers. The order of the test matrices ranges from 450 to 1700. Results from a set of numerical experiments are presented and comparisons with preconditioned GCR methods and with direct method are carried out.

A comparison of direct and preconditioned iterative techniques for sparse, unsymmetric systems of linear equations

AbstractIn this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric, sparse systems of linear algebraic equations. These problems occur in finite difference and finite element simulations of semiconductor devices, and fluid flow problems.We consider five iterative methods that appear to be the most promising for this class of problems: the biconjugate gradient method, the conjugate gradient squared method, the generalized minimal residual method, the generalized conjugate residual method and the method of orthogonal minimization. Each of these methods was tested using similar preconditioning (incomplete LU factorization) on a set of large, sparse matrices arising from finite element simulation of semiconductor devices. Results are shown where we compare the computation time and memory requirements for each of these methods against one another, as well as against a direct method that uses LU factorization to solve these problems.The results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large, sparse systems of equations, and can offer significant savings in storage and CPU time.

A Local Convergence Theory for Combined Inexact-Newton/Finite-Difference Projection Methods

Finite-difference versions of some recently developed Krylov subspace projection methods are presented and analysed in the context of solving systems of nonlinear equations using Inexact-Newton Methods. The specific projection methods considered are Arnoldi’s Method, the Generalized Minimum Residual Method (GMRES), and the Generalized Conjugate Residual Method (GCR). A local convergence theory is given for the combined Inexact-Newton/Finite-Difference Projection Methods, and the main tool used in the convergence proofs relates the results of the finite-difference method to those obtained by applying the regular projection method to a perturbed problem. In addition, several results are given which show that, under certain mild restrictions, the approximate Newton steps computed by any of the above projection methods are descent directions for the full nonlinear problem.