On Vectorizing Incomplete Factorization and SSOR Preconditioners

Computer Solution of Large Linear Systems

In this letter, an effective symmetric successive overrelaxation (SSOR) preconditioning scheme is applied to the conjugate-gradient method for solving a large system of linear equations resulting from the use of the edge-based finite-element method (FEM). With SSOR as the preconditioner as well as its efficient implementation in the conjugate-gradient (CG) algorithm, the PCG method converges five times as fast as the CG method. This result demonstrates that SSOR is a good preconditioner for the CG iterative method when the FEM is applied to solve large-scale electromagnetic problems. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 235–238, 2000.

This paper presents the results of the Connection Machine implementation of a number of preconditioners for the preconditioned conjugate gradient method. The preconditioners implemented include those based on the incomplete LU factorization, the modified incomplete LU factorization, the symmetric successive overrelaxation, and others such as several polynomial preconditioners and the hierarchical basis preconditioner. Results based on numerical experiments show that both the degree of parallelism inherent in a preconditioner and its convergence rate improvement play important roles on the overall execution time performance on parallel computers. Factors that affect the performance of the preconditioners will also be discussed. We conclude that to search for the best preconditioner on a parallel machine, we have to consider the tradeoffs between fast convergence rate and high degree of parallelism as well as the architecture of the target parallel computer.

Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equation

CHAPTER 7 - Iterative Solution of Finite Element Equations

Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some fill-in during the incomplete factorization can significantly reduce the number of iterations needed for convergence. Allowing fill-in, however, increases the time for the factorization and for the triangular system solutions. Additionally, it is difficult to predict a priori how much fill-in to allow and how to allow it. The unpredictability of the required storage/work and the unknown benefits of the additional fill-in make such strategies impractical to use in many situations. In this article we motivate, and then present, two “black-box” strategies that significantly increase the effectiveness of incomplete Cholesky factorization as a preconditioner. These strategies require no parameters from the user and do not increase the cost of the triangular system solutions. Efficient implementations for these algorithms are described. These algorithms are shown to be successful for a variety of problems from the Harwell-Boeing sparse matrix collection.

A partial preconditioned conjugate gradient method for large eigenproblems

The multifrontal method is applied for solving a large system of linear equations resulting from the use of edge-based finite element method (FEM). The finite element method combined with the perfectly matched layers (PML) is given for simulation of microwave filters and the algorithm of multifrontal method is described in detail. The reflection and insertion losses of millimeter wave filter are analyzed as the examples and the obtained results are compared with those obtained from literature. In order to demonstrate the efficiency of the multifrontal method, the computational time is compared with that of both symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (PCG) and conjugate gradient methods (CG) for the thick-iris waveguide bandpass filer.

With an increasing number of antennas and users, the complexity increases dramatically, in order to simplify the process. In this paper, We propose existed algorithms discussing their principles and their advantages as well as the comparison between each method and then we provide the possible improvement in theory changing steps in iteration process to avoid the matrix inversion, by introducing a pre-conditioned gradient (PCG) method and further improved Jacobi method and using Neumann-series terms to estimate an approximate matrix inversion and the Gauss-Seidel (GS) method with a diagonal-approximate initial solution to the method and biconjugate gradient stabilized (BICGSTAB) method and above all of the methods are based on the minimum mean square error (MMSE) detector including the application in multi-user MIMO system. Another approach is from a signal detector based on symmetric successive over relaxation (SSOR) method without matrix inversion and Neumann series based on the zero forcing signal detector. The third algorithm uses a modified version of the conjugate gradient least square (CGLS). These methods are mainly replacing the inversion process into iteration process to reduce the complexity of the algorithm not only replacing the stages but optimizing the speed from setting the initial solution as well as the improvement using the Neumann-series to replace the iteration part. Finally, we proposed a symmetric successive overrelaxation (SSOR) based Gauss-Seidel for massive MIMO detection.

The focus of this study is on the efficient solution for three-dimensional (3D) time-harmonic eddy current problems discretised by the finite-element method (FEM) with thin elements. The systems of equations arising from the finite-element formulations are solved by the geometric multigrid preconditioned conjugate gradient (MGCG) method. Numerical examples show that the MGCG method has stable convergence, which does not deteriorate for elements with high aspect ratios and its computation time is much less than that of the conjugate gradient method with incomplete Cholesky factorisation as the preconditioner (ICCG).

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

The temperature stress due to temperature changes is often more than the concrete tensile strength during construction period and operation period in massive concrete structures in hydraulic engineering, so the analysis of temperature stress, temperature control and measures to prevent the cracks become very important. The solution of large sparse linear equations is essential to finite element simulation in simulation analysis of temperature field. However, the traditional direct method will cost too much memory and calculation time, and the nodes is limited. The symmetric successive over relaxation (SSOR)-preconditioned conjugate gradient (PCG) method is a very efficient iterative method for solving large sparse linear equations. Construction process simulation and temperature filed calculation during construction period about Caohe river landing rectangle aqueduct of Beijing-Shijiazhuang emergency water supply engineering of middle-route of south-to-north water transfer project are carried out by FEM code using SSOR-PCG method, then the calculation result and monitoring value are compared. The comparison result shows that temperature field during construction can be accurately simulated by simulation calculation, which is helpful for guiding construction and choosing construction scheme, and worthy to popularize to other similar projects.

Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In the case of a finite element solution of a non-linear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver.

Application of two preconditioned generalizedconjugate gradient methods to three-dimensional neutron and photon transport equations

Parallel preconditioned conjugate gradient algorithm on GPU